GIFT  OF 
Dr.   Horace   Ivie 


Digitized  by  the  internet  Arcliive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofgeometOOdavirich 


ELEMENTS  OF  GEOMETRY 


TRIGONOMETRY, 


APPLICATIONS  IN  M^^^lJRAl'f^^^^ 


BY  CHARLES  DAVIES.  LL.  D. 

AOmOU  OF     FIRST     LESSONS    IN    ARITHMETIC,     ELEMENTARY    ALGEBRA, 
FttACTICAL    MATHEMATICS    FOR    PRACTICAL    MEN,    ELEMENTS   OF 
SURVEYING,    ELEMENTS   OF    DESCRIPTIVE    GEOMETRY, 
SHADES,    SHADOWS,    AND    PERSPECTIVE,    ANA- 
LYTICAL   GEOMETRY,    DIFFERENTIAL 
AND     INTEGRAL    CALCULUa 


A.    S.    BARNES    &   COMPANY, 
NEW  YORK  AND  CHICAGO. 


GIFTOF     _     ,  QAh-Z9 

DAVIES' MTHEMATICS.  rl!^ 

And  Only  Thorough  and  Complete  Mathematical  Series. 


ii<r    tp3:e,eb   tp^^^-^its,. 


I.    COMMON   SCHOOL    COURSE 

Davies'  JPrimary  Arithmetic— The  fundamental  principles  dieplayed  in 

the  Object  Lessons. 
Davies'  Intellectual  Arithmetic. — Referring  all  operations  to  the  unit  1  as 

the  only  tangible  basis  for  logical  development. 
Davies'  ISlernents  of  Written  Arithmetic. A  pmctical   introduction 

to  the  whole  subject.    Theory  subordinated  to  Practice. 
Davies'   Practical  Arithmetic*  —  The  most  successful   combination   of 

Theory  and  Practice,  clear,  exact,  brief,  and  comprehensive. 

.,   II.  JOAOeM/C  COURSE. 

Davies'  JJniversity  'Arithtnetic* — Treating  the  subject  exhaustively  as 

a  sde:4/;e,iii  a  Ipglcal  series  of  connected  propositions. 
Uames'  :l^}lemyeitUu'i'i)  A^ehra.*r-A  connecting  link,  conducting  the  pupil 

easily  from  afithnleticai  {)ro'cc6<^ps"to  abstract  analysis. 
Davies'  University/  Alf/ebra.*— For  institutions  desiring  a  more  complete 

but  not  the  fullest  course  in  pure  Algebra. 
Davies'  l^ractical  3Iathetn<ttics. — The  science  practically  applied  to  the 

useful  arts,  as  Drawing,  Architecture,  Sur^'eying,  Mechanics,  etc. 
Davies'  Elementary  Gf^owie/j'?/.— The  important  principles  in  simple  form, 

but  with  all  the  exactness  of  vigorous  reasoning. 
Davies'  Elements  of  Surveying.— Re-wvMian  in  1870.     The  simplest  and 

most  practical  presentation  for  youths  of  12  to  16. 

///.  COLLEGIATE  COURSE. 

Davies'  Bourdon's  .4 Zr/e bra.*— Embracing  Sturm's  Theorem,  and  a  most 
exhaustive  and  scholarly  course. 

Davies'   XTniversity  Algebra.*— K%hori(^.v  course  than  Bourdon,  for  Insti- 
tutions have  less  time  to  give  the  subject. 

Davies'  Legendre's  Geometry.— AcknowleigeA  the  o/i?y  satisfactory  trea- 
tise of  its  grade.    .300,000  copies  have  been  sola. 

Davies'  Analytical    Geometry   and   Calculus. — The  shorter  treatises, 
comoined  in  one  volume,  are  more  available  for  American  courses  of  study. 

Davies'  Analytical  Geometry.  |The  original  compeudiums,  for  those  de- 

Davies'  Diff.  &  Int.   Calculus.  J      siring  to  give  full  time  to  each  branch. 

Davies'  Descriptive  Geometry. — With  application  to  Spherical  Trigonome- 
try, Spherical  Projections,  and  Warped  Surfaces. 

Davies'  Shades,  Shadows,  and.  I*erspective.—A  succinct  exposition  of 
the  mathematical  principles  involved. 

Davies'  Science  of  Mathematics. — For  teachers,  embracing 

I.  Grammar  of  Arithmetic,       III.  Logic  and  Utility  op  Mathematics, 
II.  Outlines  of  Mathematics,    IV.  Mathematical  Dictionary. 


*  Keys  may  be  obtained  from  the  Publishers  by  Teachers  only. 

Copyright,  1858,  by  Charles  Davies. 
Copyright  Renewed,  1883,  by  Mary  Ann  Davies. 
EL.  GEOM.  GDUCATfON  DEFT 


PREFACE 


Those  who  are  conversant  with  the  prcpcj-ation  of  ele- 
mentary text-books,  have  experienced  the  difficuhy  of 
adapting  them  to  the  various  wants  which  they  are  in- 
tended to  supply. 

The  institutions  of  education  are  of  all  grades,  from  the 
college  to  the  district  school,  and  although  there  is  a  wide 
difference  between  the  extremes,  the  level,  in  passing 
from  one  grade  to  the  other,  is  scarcely  broken. 

Each  of  these  classes  of  seminaries  requires  text-books 
adapted  to  its  own  peculiar  wants ;  and  if  each  held  its 
proper  place  in  its  own  class,  the  task  of  supplying  suit- 
able works  would  not  be  difficult. 

An  indifferent  college  is  generally  inferior,  in  the  system 
and  scope  of  its  instruction,to  the  academy  or  high  school; 
while  the  district  school  is  often  found  to  be  superior  to 
its  neighboring  academy. 

The  Geometry  of  Legendre,  embracing  a  complete 
course  of  Geometrical  science,  is  all  that  is  desired  in 
the  colleges  and  higher  seminaries ;  while  the  Practical 
Mathematics  for  Practical  Men,  recently  published,  is 
designed  to  meet  the  wants  of  those  schools  which  are 
strictly  elementary  and  practical  in  their  systems  of 
instruction.  i , 

924173 


6  PREFACE. 

But  still  a  krge  class  of  seminaries  remained  uusup- 
plied  with  a  suitable  text-book  on  Elementary  Geometry 
and  Trigonometry  :  viz.,  tliose  where  the  pupils  are  car- 
ried beyond  the  acquisition  of  facts  and  mere  practicaJ 
knowledge,  but  have  not  time  to  go  through  with  a  full 
course  of  mathematical  studies. 

It  is  tor  such,  that  the  following  work  is  designed.  It 
has  been  the  aim  of  the  author  to  present  the  striking 
and  important  triiths  of  Geometry  in  a  form  more  simple 
and  concise  than  could  be  adopted  in  a  complete  treatise 
and  yet  to  preserve  the  exactness  of  rigorous  reasoning. 

In  this  system  of  Geometry  nothing  has  been  taken  fo^ 
granted,  and  nothing  passed  over  without  being  fully  de 
monstrated. 

The  Trigonometry,  including  the  applications  to  the 
measurements  of  heights  and  distances,  has  been  writ- 
ten upon  the  same  plan  and  for  the  same  objects:  it 
embraces  all  the  important  theorems  and  all  the  striking 
examples. 

In  order,  however,  to  render  the  applications  of  Ge- 
ometry to  the  mensuration  of  surfaces  and  solids  complete 
in  itself,  a  few  rules  have  been  given  which  are  not  de- 
monstrated. This  forms  an  exception  to  the  general  plan 
of  the  work,  but  being  added  in  the  form  of  an  appendix,  it 
does  not  materially  break  its  unity. 

That  the  work  may  be  useful  in  advancing  the  interestg 
of  education,  is  the  hope  and  ardent  wish  of  the  author. 
FisiiEiLL  Landing, 
May.  1861 


CONTENTS. 

BOOK  I. 

DBTimTioNs  and  Remarks, 0—10 

Axioms,  ----  ...  iQ 

Properties  of  Polygons,    --.---  17 — 87 

BOOK   II. 

Of  the  Circle, 88 

Problems  relating  to  the  First  and  Second  Books,        -  -         53-  -4i8 

BOOK  III. 

Ratios  and  Proportions,     -        -  -  .        6{>-^l 

BOOK  IV. 

Measurement  of  Areas  and  Proportions  of  Figures,      -  •       82 — 108 

PrDblcms  relating  to  the  Fourth  Book,        .        .         -  .     109—113 

Appendix — Regular  Polygons,  -        -         -  .     113 — US 

BOOK  V. 
Of  Planea  and  their  Angles,      -        •        -  110^120 

BOOK    VI 

O!  Solids,  -     126—162 

Appendix, -     IG3— 164 


CONTENTS. 
TRTGONOMETRT. 


Or  LoGARiroMa, 

Of  Scales, 

Definitions,  and  Exjlanation  of  TablevB,  - 

llieorcms, 

Examples, 

Application  to  Heights  and  Distances,     - 

APPLICATIONS  OF  GEOMETRY. 

MeNSUBATION    of    SlTRFACEB, 

General  Principles, 
Contents  of  Figures, 

Mensuration  of  Solids, 
General  Principles, 
Solidities  of  Figures, 

Mensuration  of  the  Round  Bodiis, 
To  find  the  Surface  of  a  Cylinder, 
To  find  the  Solidity  of  a  Cylinder, 
To  find  the  Suiface  of  a  Cone, 
To  find  the  Solidity  of  a  Cone, 
To  find  the  Surface  of  the  Frustum  of  'a  Cone, 
To  find  the  Solidity  of  the  Frustum  of  a  Cono, 
To  find  the  Surface  of  a  Sphere,    - 
To  find  the  Surface  of  a  Spherical  Zone, 
To  find  the  Solidity  of  a  Sphere,    - 
To  find  the  Solidity  of  a  Spherical  Segment, 
To  find  the  Solidity  of  a  Spheroid, 
T::  find  the  Surface  of  a  Cylindrical  Ring, 
To  find  the  Solidity  of  a  Cylindrical  Ring, 


Pace. 
166—17(1 
176—181 
181—189 
189—193 
193—201 
202— 21  n 


211 

211-213 

218—239 

289 

289—240 
240-241 

248 

248—249 

249—260 

250—261 

251--262 

253 

264 

266 

266—266 

266—267 

268 

269—260 

260-261 

261—208 


ELEMENTARY 


BOOK   I. 


DEFINITIONS    AND    REMARKS. 

1.  Extension  has  three  dimensions,  length,  breadth,  anJ 
thickness. 

Geometry  is  the  science  which  has  for  its  object: 

Ist.  The  measurement  of  extension ;  and  2dly,  To  discover, 

by  means   of  such  measurement,  the  properties  and  relationa 

of  geometrical  figures. 

2.  A  Point  is  that  which  has  place,  or  position,  but  not 
magnitude. 

3.  A  Line  is  length,  without   breadth  or  thickness. 

4.  A    Straight    Line    is   one    which   lies 

in   the   same  direction   between  any   two  of    — — — — 
its  points. 

6.  A   Curve  Line  is   one   which  changes 
is  direction    at   every  point. 

The  word  lirie  when  used  alone,  will  designate  a  straight 
line  ;   and   the   word   curve^  a  curve  line. 

6.  A  Surface  is  that  which  has  length  and  breadth,  with- 
out height  or  thickness. 

7.  A  Plane  Surface  is  that  which  lies  even  throughout  its 
whole  extent,  and  with  which  a  straight  line,  laid  in  nny 
direction,  will  exactly  coincide  in  its  whole  lensfth. 

8.  A  Curved  Surface  has  length  and  breadth  without  thick- 
Dees,  and  like  a  curve  line  is  constantly  changing  its  direction 

9.  A  Solid  or  Body  is  that- which  has  length,  breadth,  and 
ihickness.     Length,  breadth,  and  thickness  are  called  dimcn- 


10  GEOMETRY. 


IDefinitior  s. 


eiaas.;    II'«rtc|e^.  ^"  feC'lid;  has  three  dimensions,  a  surface  two 
and  a  line  one.     A  poiiiv'  lias.  Ho  dimensions,  but  position  only 

10.  Geometry  treats  of  lines,  surfaces,  and  solids. 

11.  A  Demonstration  is  a  course  of  reasoning  which  estalv 
hshcs  a  truth. 

12.  An  Hypothesis  is  a  supposition  on  which  a  demonstra' 
tion  may  be  founded. 

13.  A  Theorem  is  something  to  be  proved  by  demonstration. 

14.  A  Problem  is  something  proposed  to  be  done. 

15.  A  Proposition  is  something  proposed  either  to  be  done 
or  demonstrated — and  may  be  either  a  problem  or  a  theorem. 

16.  A  Corollary  is  an  obvious  consequence,  deduced  from 
something  that  has  gone  before. 

17.  A  Scholium  is  a  remark  on  one  or  more  preceding  propo- 
sitions. 

18.  An  Axiom  is  a  self  evident  proposition. 

OF   ANGLES. 

19.  An  Angle  is  the  portion  of  a  plane  included  between 
two  straight  lines  which  meet  at  a  common  point.  The 
two  straight  lines  are  called  the  sides  of  the  angle,  and 
the  common   point   of  intersection,   the   vertex. 

Thus,   the   part  of  the   plane   included  P 

between  AB  and  AC  is  called  an  angle :  ^^ 

AB  and  A  C  are  its  sides^  and  A  its  vertex.     J^  fn 

An  angle  is  generally  read,  by  placing  the  letter  at  the  vci 
tox  in  the  middle.  Thus,  we  say,  the  angle  CAB.  We  may 
however,  say  simply,  the  angle  A. 

20.  One  line  is  said  to  be  perpendicular  to  andther  when  it 


BOOK     I 


II 


Definitions 


B  C 


J) 


B C 


The  two  angles  formed  are  then  equal  to 
each  other.  Thus,  if  the  line  DB  is  per- 
pendicular to  AC,  the  angle  DBA  will 
be  equal  to  DEC. 

21.  When  two  lines  are  perpendicular 
to  each  other,  the  angles  which  they  form 
are  called  right  angles.  Thus,  DBA  and 
DBC  RTG  called  right  angles. 

22.  An  acute  angle  is  less  than  a  right 
angle.     Thus,  DBC  is  an  acute  angle. 

23.  An  obtuse  angle  is  greater  than  a 
right  angle.  Thus,  DBC  is  an  obtuse 
angle. 

24.  The  circumference  of  a  circle  is  a 
curve  line  all  the  points  of  which  arc 
equally  distant  from  a  certain  point  within 
called  the  centre. 

Thus,  if  all  the  points  of  the  curve  A£B 
are  equally  distant  from  the  centre  C,  this 
curve  will  be  the  circumference  of  a  circle. 

25.  Any  portion  of  the  circumference, 
OS  AED,  is  called  an  arc 

26.  The  diameter  of  a  circle  is  a 
straight  line  passing  through  the  centre 
and  terminating  at  the  circumference. 
Thus,  A  CB  is  a  diameter. 

27.  One  half  of  the  circum^'erence,  as 
ACB  is  called  a  semicircumference ;  and 
one  quarter  of  the  circumference,  as  -4C 
IS  called  a  quadrant 


12 


GEOMETRY. 


Definitions. 


28.  The  circumference  of  a  circle  is  used  for  the  measuro« 
ment  of  angles.  For  this  purpose  it  is  divided  into  360  equal 
parts  called  degrees,  each  degree  into  60  equal  parts  called 
minutes,  and  each  minute  into  60  equal  parts  called  seconds. 
The  degrees,  minutes,  and  seconds  are  marked  thus  "  '  " ;  and 
9**  18'  16",  are  read,  9  degrees  18  minutes  and  16  seconds. 

29.  Let  us  suppose  the  circumference 
of  a  circle  to  be  divided  into  360  degrees, 
beginning  at  the  point  B.  If  through 
the  point  of  division  marked  40,  we  draw 
CjE,  then,  the  angle  E  CB  will  be  equal  to 
40  degrees.  If  CF  were  drawn  through 
the  point  of  division  marked  80,  the  angle  BCF  would  be  e(|ual 
to  80  degrees. 


OF    LINES. 

30.  Two  straight  lines  are  said  to  be 
parallel^  when  being  produced  either  way, 
as  far  as  we  please,  they  will  not  meet 
each  other. 

31.  Two  curves  are  said  to  be  parallel 
or  concentric,  when  they  are  the  same  dis- 
tance from  each  other  at  every  point. 

32.  Oblique  lines  are  those  which  ap- 
proach each  other,  and  meet  if  sufficiently 
produced. 

33.  Lines  which  are  parallel  to  the  horizon,  or  to  the  water 
level,  are  called  hor'zontal  lines. 

34.  Lines  which  are  perpendicular  to  the  horizon,  or  to  the 
wafer  level  are  cl^Ued  vertical  lines 


B  O  O  K     1  .  13 


Defin  it  ions, 


or    PLANE    FIGURES. 

35.  A  Plane  Figure  is  a  portion  of  a  plane  terminated  on  a]1 
9ide8  by  lines,  either  straight  or  curved. 

36.  If  the  lines  which  bound  a  figure  are  straight,  the  space 
wliich  they  inclose  is  called  a  rectilineal  figure,  oi  polygon. 
The  lines  themselves,  taken  together,  are  called  the  perinuter 
of  the  polygon.  Hence,  the  perimeter  of  a  polygon  is  the  sum 
of  all  its  sides. 


37.  A  polygon  of  three  sides  is  called 
a  triangle. 


38.  A  polygon  of  four  sides  is  called 
a  quadrilateral. 


39.  A  polygon  of  five  sides  is  called  a 
pentagon. 


40.  A  polygon  of  six  sides  is  called 
hexagon. 


41.  A  polygon  ot  seven  sides  is  called  a  heptagon 

42    A  polygon  of  eight  sides  is  called  an  octagon. 
3 


14 


GEOMETRY 


De  f  init  ions. 


43.  A  polygon  of  nine  sides  is  called  a  nonagon. 

44.  A  polygon  of  ten  sides  is  called  a  decagon. 

45.  A  polygon  of  twelve  sides  is  called  a  dodocagon. 

46.  There  are  several  kinds  of  triangles. 


First.  An  equilateral  triangle,  which  has 
its  three  sides  all  equal. 


Second.  An  isosceles  triangle,  which  has 
two  of  its  sides  equal. 


Third.  A  scalene  triangle,  which  has  its 
three  sides  all  unequal. 


Fourth.  A  right  angled  triangle,  which 
has  one  right  angle. 

In  the  right  angled  triangle  ABCy  the 
side  A  C,  opposite  the  right  angle,  is  called 
the  hypothenuse. 

47.  The  base  of  a  triangle  is  the  side  on 
which  it  stands.  Thus,  AB  is  the  base  of 
the  triangle  ACB. 

The  altitude  of  a  triangle  is  a  line  drawn 
from  the  angle  opposite  the  baae  and  per-^ 
pendicular  to  the  base.     Thus,  CD  is  the  altitude  of  the  tri 
angle  ACB 


BOOK    I 


u 


Definitions. 


48.  There  are  three  kinds  of  quadrilaterftla. 


1.  The  trapezium^  which  has  none  of 
)tB  sides  parallel. 


2.  The  trapezoid^  which  has  only  two 
of  its  sides  parallel. 


mx 


8.  The  parallelogram^  which- has  its 
opposite  sides  parallel. 


7 


4y.  There  are  four  kinds  of  parallelograms 

1.  The  rhomboid^  which  has  no  right 
angle. 


2.  The  rhombus^  or  lozenge^  which  is 
an  equilateral  rhomboid. 


8.  The  rectangle^  which  is  an  equian- 
galar  parallelogram. 


4.  The  square^  which  is  both  equilat- 
eral and  equiangular. 


16  GEOMETRY. 


Of  Axioms. 


60.  A  Diagonal  of  a  figure  is  a  line  which 
joins  the  vertices  of  two  angles  not  adjacent. 


61.  The  base  of  a  figure  is  the  side  on  which  it  is  supposed 
U)  stand ;  and  the  altitude  is  a  line  drawn  from  the  opposite 
side  or  angle,  perpendicular  to  the  base, 

AXIOMS. 

1.  Things  which  are  equal  to  the  same  thing  are  equal  to 
each  other. 

2.  If  equals  be  added  to  equals,  the  wholes  will  be  equal. 

3.  If  equals  be  taken  from  equals,  the  remainders  will  be 
equal. 

4.  If  equals  be  added  to  unequals,  the  wholes  will  be  un- 
equal. 

5.  If  equals  be  taken  from  unequals,  the  remainders  will  be 
unequal. 

6.  Things  which  are  double  of  equal  things,  are  equal  to 
each  other. 

7.  Things  which  are  halves  of  the  same  thing,  are  equal  to 
each  other. 

8.  The  whole  is  greater  than  any  of  its  parts 

9.  The  whole  is  equal  to  the  sum  of  all  its  parts. 

10.  All  right  angles  are  equal  to  each  other. 

11.  A  straight  line  is  the  shortest  distance  between  two 
points. 

12.  Magnitudes,  which  being  applied  to  each  other,  ooin- 
ddo  throughout  their  whole  extent,  are  equal. 


BOOK     1  . 


17 


Of  Angles 


PROPERTIES    OF    POLYGONS. 


THEOREM    I. 

Every  diameter   of  a  circle  divides  the   circumference  into   tvxy 
equal  parts. 

Let  ADBE  be  the  circumference  of  a 
circle,  and  A  CB  a  diameter :  then  will 
the  part^Di?  be  equal  to  the  part  AEB. 

For,  suppose  the  part  AEB  to  be  turn- 
ed around  AB,  until  it  shall  fall  on  the 
part  ADB.  The  curve  AEB  will  then 
exactly  '  oincide  with  the  curve  ADB,  or  else  there  would 
be  some  point  in  the  curve  AEB  or  ^ Z)5,  unequally  distant 
from  the  centre  C,  wliich  is  contrary  to  the  definition  ot  a 
circumference  (Def.  21).  Hence,  the  two  curves  will  bo 
equal  (Ax.  13). 

Corollary  1.  If  two  lilies,  AB,  DE, 
be  drawn  through  the  centre  C  perpen- 
dicular to  each  other,  each  will  divide  the 
circumference  into  two  equal  parts ;  and 
the  entire  circumference  will  be  divided 
into  the  equal  quadrants  DB,  DA.  AE, 
and  EB. 

Cor.  2.  Hence,  a  right  angle,  as  DCB^  is  measured  by  one 
quadrant,  or  90  degrees;  two  right  angles  by  a  semicircumfer* 
encc,  or  130  degrees ;  and  four  right  angles  by  the  whole  cLr« 
cumfercnce,  or  360  degrees 


18 


U  E  0  M  E  T  R  V  . 


Of  Angles. 


D 


THEOREM    II. 

If  one  straight  line  meet  another  straight  linc^  the  sum  of  thA 

two  adjacent  angles  will  he  equal  to  two  right  angles. 

[iCt  the  straight  line  CD  meet  the 
straight  line  AB^  at  the  point  C;  then 
will  the  angle  DCB\>\\xs  the  angle  DC  A 
be  equal  to  two  right  angles-  A  C  B 

About  the  centre  C,  with  any  radius  as  CB,  suppose  & 
semicircumference  to  be  described.  Then,  the  angle  DCB 
will  be  measured  by  the  arc  BD,  and  the  angle  DC  A  by  the 
arc  AD.  But  the  sum  of  the  two  arcs  is  equal  to  a  seraicir- 
cumference  •  hence,  the  sum  of  the  two  angles  is  equr.l  to  two 
right  angles  (Th.  i,  Cor.  2). 

Co7.  1.  If  one  of  the  angles,  as  DCB, 
is  a  right  angle,  the  other  angle,  DC  A 
will  also  be  a  right  angle. 

Cor.  2.  Hence,  all  the  angles  which 
can  be  formed  at  any  point  (7,  by  any 
number  of  lines,  CD,  CE,  CF,  &c., 
drawn  on  the  same  side  of  AB,  are  equal 
to  two  right  angles :  for,  they  will  be 
measured  by  a  semicircumference. 

Cor.  3.  li  DC  meets  two  lines  CB,  CA,  making  DCB 
plus  DCA  equal  to  two  right  angles,  ACB  will  form  ohp 
straight  line. 

Cor.  4.  Hence,  also,  all  the  anglea 
which  can  be  formed  round  any  point,  as 
(7,  are  equal  to  four  right  angles.  For, 
the  Bum  of  all  the  arcs  which  measure 
them,  is  equal  to  the  entire  circumference, 
w-hich  is  the  measure  of  four  right  angles   (Th.  i.  Cor.  2). 


B  0  O  K     1  .  19 


Of  Triangles 


THEOREM    III. 

Ij  ivDO  Straight  lines  intersect  each  other^  the  opposite  or  ocr- 
tical  angles  which  they  form ^  are  equal. 

Let  ihc  two  straight  lines  AB  and 
CD  intersect  each  other  at  the  point 
E :  then  will  the  opposite  angle  A  EC 
be  equal  to  Z)£B,  and  AED=zCEB. 

For,  since  the  line  AE  meets  the  "c 

line  CD,  the  angle  AEC-{-AED:=  two  right  angles.  But 
since  the  line  DE  meets  the  line  AB,  we  have  DEB-{-AED= 
two  right  angles.  Taking  away  from  these  equals  the  com- 
mon angle  AED,  and  there  will  remain  the  angle  AEC  equal 
to  the  angle  DEB  (Ax.  3). 

In  the  same  manner  we  may  prove  that  the  angle  AED  is 
equal  to  the  angle  CEB. 

THEOREM   IV. 

Jf  two  triangles  hctve  two  sides  and  the  included  angle  of  the 
one,  equal  to  two  sides  and  the  included  angle  of  the  other,  each 
to  each,  the  two  triangles  will  be  equal. 

Let  the  triangles  ABC  and  DEF 
have  the  side  AC  equal  to  DF,  CB 
to  FE,  and  the  angle  C  equal  to  the 
angle  F :  then  will  the  triangle  A  CB 
be  equal  to  the  triangle  DEF. 

For,  suppose  the  side  ^4  C,  of  the     -^  ^  ^  ^ 

triangle  ACB,  to  be  placed  on  DF,  so  that  the  extremity  C 
Bhall  fall  on  the  extremity  F:  then,  since  the  sides  are  equaj 
A  will  fall  on  D. 

But  since  the  angle  C  is  equal  to  the  angle  F,  the  line  CB 


20  GEOMETRY. 


Of    TriangIo8 


will  fall  on  FE ;  and  since  CB  is  equal 
to  FE,  the  extremity^  will  fall  on  E ; 
and  consequently  the  side  AB  will  fall 
on  the  side  DE  (Ax.  11).  Hence,  the 
two  triangles  will  fill  the  same  space, 
and  consequently  are  equal  (Ax.  12.).     ^  ^  F> 

Scholium.  Two  triangles  are  said  to  be  equal,  when  being 
applied  the  one  to  the  other  they  exactly  coincide  (Ax.  12). 
Ilence,  equal  triangles  have  their  like  parts  equal,  each  to 
each,  since  those  parts  coincide  with  each  other.  The  converse 
of  the  proposition  is  also  true,  namely,  that  two  triangles 
which  have  all  the  parts  of  the  one  equal  to  the  corresponding 
"parts  of  the  other^  each  to  each,  are  equal :  for  if  applied  the 
one  to  the  other,  the  equal  parts  will  coincide. 

THEOREM    V. 

If  two  triangles  have  two  angles  and  the  included  side  of  tnt 
one,  equal  to  two  angles  and  the  included  side  of  the  other,  each  to 
rach,  the  two  triangles  will  be  equal. 

Let  the  two  triangles  ABC  and 
DEF  have  the  angle  A  equal  to  the 
angle  D,  the  angle  B  equal  to  the 
angle  E,  and  the  included  side  AB 
equal  to  the  included  side  DE  •  then 
will  the  triangle  ABC  bo  equal  to  the 
triangle  DEF. 

For,  let  the  side  AB  be  placed  on  the  side  DE,  the  cxtrem 

ity  A  on  the  extremity  D ;  and  since  the  sides  are  equal,  the 
point  B  will  fall  on  the  point  E. 

Then  since  <he  angle  A  is  equal  to  the  angle  D,  the  sidr 


O  O  K     1  .  5^1 


0  f    T  r  i  a  n  g  1  o  a 


AC  will  take  the  direction  DF :  and  since  the  angle  B  u 
equal  to  the  angle  jE,  the  side  BC  will  fall  or  the  side  EF : 
hence,  the  point  C  will  be  found  at  the  same  time  on  DF  and 
EF,  and  therefore  will  fall  at  the  intersection  F:  consequcrlly, 
all  the  parts  of  the  triangle  ABC  will  coincide  with  tiie  parti! 
of  the  triangle  DEF,  and  therefore,  the  two  triangles  arc  equal 

THEOREM    VI. 

In  an  isosceles  triangle  the  angles  opposite  the  equal  sides  are 
equal  to  each  other. 

Let  ABC  be  an  isosceles  triangle,  liav- 
ing  the  side  A  C  equal  to  the  side    CB : 
then  will  the  angle  A  bo  ec^ual  to  the  an- 
gle B. 
^  AD  B 

For,  suppose  the  line  CD  to  be  drawn  dividing  the  angle  C 
into  two  equal  parts. 

Then,  the  two  triangles  ACD  and  DCB,  have  two  sides  and 
ihe  included  angle  of  the  one  equal  to  two  sides  and  the  in- 
cluded angle  of  the  other,  each  to  each :  that  is,  the  side  A  C 
mjual  to  BC,  the  side  CD  common,  and  the  included  angle 
ACD  equal  to  the  included  angle  DCB :  hence  the  two  trian 
glcs  are  equal  (Th.  iv) ;  and  hence,lhe  angle  A  is  equal  to 
the  angle  B. 

Cor.  1.  Hence,  the  line  which  bisects  the  vertical  angle  of 
an  isosceles  triangle,  bisects  the  base.  It  is  also  perpendicu- 
lar to  the  base,  since  the  angle  CD  A  is  equal  to  the  angle 
CDB. 

Cor.  2.  Hence,  also,  every  equilateral  triangle,  must  also 
be  equiangular:  that  is,  have  all  its  angles  equal,  each  to  each 


22  GEOMETRY 


Of    Triangles 


THEOREM    VII. 

Conversely. — If  a  triangle  has  two  of  its  angles  equals  iJu 
sides  opposite  those  angles  will  also  be  equal. 

In  the  triangle  ABC,  let  the  angle  A  be 
equal  to  the  angle  B :  tnen  will  the  side 
BO  he  equal  to  the  side  AC. 

For.  if  tlie  two  sides  are  not  equal,  one 
of  them  must  be  greater  than  the  other. 
Suppose  ^  C  to  be  the  greater  side.  Then 
take  a  part  AD  equal  to  BC 

Now,  in  the  two  triangles  ADB  and  ABC,  we  have  the 
side  AD  =  BC,  by  hypothesis •,  the  side  aB  common,  and  the 
angle  A  equal  to  the  angle  B :  hence,  the  two  triangles  have 
two  sides  and  the  included  angle  of  the  one  equal  to  two  sides 
and  the  included  angle  of  the  other,  each  to  each :  hence,  the 
two  triangles  are  equal  (Th.  iv),  that  is,  a  part  ADB  is 
equal  to  the  whole  ABC,  which  is  impossible  (Ax.  8) :  conse- 
quently, the  side  AC  cannot  be  greater  than  the  side  CB,  and 
hence,  the  triangle  is  isosceles. 

Scholium  1.  The  method  of  reasoning  pursued  in  the  last 
theorem,  is  called  the  "  reductio  ad  absurdum,"  or  a  proof  that 
leads  to  a  known  absurdity. 

Let  us  analyze  this  method  of  reasoning.  We  wished  to 
prove  that  the  two  sides  A  C,  CB  were  equal.  We  supposed 
them  unequal,  anl  ^C  the  greater — that  was  an  hypothesis 
(See  Def.  12).  We  then  reasoned  on  the  hypothesis  and 
proved  a  part  equal  to  the  whole,  which  we  know  to  l)e  false 
(Ax.  8)  Hence,  we  conclude  that  the  hypothesis  is  untrue, 
because  after  a  correct  chain  of  reasoning  it  leads  to  a  resull 
which  we  know  to  be  absurd 


B  O  O  K    1 


23 


Of    Triangles. 


Scholium  2.  Generally, — If  the  demonstration  is  based  Of. 
known  principles,  previously  proved,  or  admitted  in  the  ax- 
ioms, the  conclusion  will  always  be  true.  But,  if  the  demon- 
stration is  based  on  an  hypothesis,  (as  in  the  last  theorem,  thai 
A  C  was  the  ^eater  side),  and  the  conclusion  is  contrary  to 
what  has  been  previously  proved,  or  admitted  in  the  axioms 
then,  it  follows,  that  the  hypothesis  cannot  be  true. 

The  former  is  called  a  direct^  and  the  latter  an  indirect 
demonstration. 


THEOREM    VIII. 

If  two  triangles  have  the  three  sides  of  the  one  equal  to  the 
three  sides  of  the  other ^  each  to  each,  the  three  angles  wiU  aha  be 
equal,  each  to  each. 

Let  the  two  triangles  ABC,  ABD, 
have  the  side  AB  equal  to  the  side  AB, 
the  side  AC  equal  to  AD,  and  the  side 
CB  equal  to  DB :  then  will  the  corres- 
ponding angles  also  be  equal,  viz :  the 
angle  A  will  be  equal  to  the  angle  A,  the 
angle  B  to  the  angle  B,  and  the  angle  C 
to  the  angle  D. 

For,  suppose  the  triangles  to  be  joined 
by  their  longest  equal  sides  ABy  and  the 
line  CD  to  be  drawn. 

Then,  since  the  side  AC  is  equal  to  AD,  by  hypothesis,  the 
mangle  ADC  will  be  isosceles;  and  therefore,  the  angle  ACD 
will  be  equal  to  the  angle  ADC  (Th.  vi).  In  like  manner, 
in  the  triangle  CBD,  the  side  CB  is  equal  to  DB  :  hence,  the 
angle  BCD  is  equal  to  the  angle  BDC. 

Now,  by  the  addition  of  equals,  we  have 


24 


GEOMETRY 


Of   Trianglea. 


ACD-\-BCD  =  ADC+BDC 
that  is,  the  angle  ACB=ADB. 

Now,  the  two  triangles  ACB  and  ADB 
have  two  sides  and  the  included  angle  of 
the  one  equal  to  two  sides  and  the  in- 
cluded angle  of  the  other,  each  to  each:  hence,  the  remainirig 
angles  will  be  equal  (Th.  iv) :  consequently,  the  angle  CAB 
is  equal  to  BAD,  and  the  angle  CBA  to  the  angle  ABD. 

Sch.  The  angles  of  the  two  triangles  which  are  equal  to 
each  other,  are  those  which  lie  opposite  the  equal  sides. 


THEOREM    IX. 

If  one   side  of  a  triangle   is  produced,  the   outward   angle  ts 

greater  than  either  of  the  inward  opposite  angles. 

Let  ABC  be  a  triangle,  having  the  side 
AB  produced  to  D  :  then  will  the  outward 
angle  CBD  be  greater  than  either  of  the 
inward  opposite  angles  A  or  C. 

For,  suppose  the  side  CB  to  be  bisected  at  the  point  E. 
Draw  AE,  and  produce  it  until  EF  is  equal  to  AE,  and  then 
draw  BF. 

Now,  since  the  two  triangles  AEC  and  BEF  hrt\e  AE=z 
EF  and  EC=EB,  and  the  included  angle  AEC  equal  to  the 
included  angle  BEF  (Th.  iii),  the  two  triangles  will  be  equal 
in  all  respects  (Th.  iv) :  hence,  the  angle  EBF  will  be  equal 
U)  the  angle  C.  But  the  angle  CBD  is  greater  than  the  angle 
CBFf  consequently  it  is  greater  than  the  angle  C, 

In  like  manner,  if  CB  be  produced  to  6r,  and  AB  be  bi- 
sected, it  may  be  proved  that  the  outward  angle  ABG,  or  its 
equal  CBD  (Th.  iii),  is  greater  than  the  angle  A. 


B  O  O  K      I.  26 

Of   T  r  i  a' II g  log. ^_^___ 


THEOREM    X. 

The    sum  of  any  two   sides  of  a  triangle  is  theater  than  the 
third  ride. 

Let  A  BC  he  a  triangle  •  tlicn  will  the 

"  C 

sum  of  two  of  its  sides,  as  A  C,  CB,  be 

g  eater  than  the  third  side  AB. 

For  tlie  straight  line  .4J5  is  the  short- 

esi  distance  between  the  two  j»oints  A  and  B  (Ax.  xi):  hence 

A  C-\-  CB  is  greater  than  AB. 

THEOREM    XI. 

Thfi  r^reater  side  of  every  triangle  is  opposite  the  greater  ang^ . 
and  conversely,  the  greater  angle  is  opposite  the  greater  side. 

First.  In  the  triangle  CAB,  let  the  an- 
gle C  be  greater  than  the  angle  B :  then, 
will  the  side  A  B  be  greater  than  the  side 
AC.  L- 


C  B 

For,  draw  CD,  making  the  angle  BCD 

equal  to  the  angle  B.  Then,  the  triangle  CBD  will  be 
isosceles:  hcnco,  the  side  CD  =  DB  (Th.  vii.) 

Hut,  by  the  last  theorem  AC  is  less  than  AD-\-CD ;  thai 
is.  less  than  AD-\-DB,  and  consequently  less  than  AB. 

Secondly.  Let  us  suppose  the  side  AB  io  be  greater  than 
A  C ;  ilicn  will  the  angle  C  be  greater  than  the  angle  B. 

For  if  ilie  angle  C  were  equal  to  B,  the  triangle  CA  /• 
would  he  isosceles,  and  the  side  AC  would  be  equal  io  AB 
{T]\.  vii),  which  would  be  contrary  to  the  hypothesis. 

Again,  if  the   angle  C  were  less  than   B,  then,  by  the  first 

part  of  the  theorem,  the   side  AB  would  be   loss   than  AC. 

which  is  also  contrary  to  tho  hypothesis      (lence.  since  C 
3 


2()  G  E  O  i>l  E  T  R  Y . 

Of    Parallel    Lines. 

cannoi  be  equal  to  B,  nor  less  than  B,  it  follows  iliat  it  musi 
bo  greater 

TIIEOREJl    XII. 

If  a  straight  line  intersect  two  parallel  lints,  the  alternate  angi:^ 
will  be  equal. 

I  f  two  parallel  straight  lines,  A  B  CD, 
are  intersected  by  a  third   line  GN,  the  p/^ 


angles  AEF  and  EFD  are  called  alternate    ^         yC-"'"     ^ 
angles.     It  is  required  to  prove  that  these    C     '/' f  D 

IT 

angles  are  equal. 

If  they  are  unequal  one  of  them  must  be  greater  than  the 
other.     Suppose  EFD  to  be  the  greater  angle. 

Now  conceive  FB  to  be  drawn,  making  the  angle  EFB 
equal  to  the  angle  AEF,  and  meeting  ^^in.  B 

Then,  in  the  triangle  FEB  the  outward  angle  FEA  is  greater 
than  either  of  the  inward  angles  B  or  EFB  (Th.  ix.) ;  and 
therefore,  EFB  can  never  be  equalto^^FsolongasFii  meets 
EB. 

But  since  we  have  supposed  EFD  to  be  greatei  than  AEF, 
it  folloAVS  tliat  EFB  could  not  be  equal  to  AEF,  if  FB  fell  be- 
low FD.  Therefore,  if  the  angle  EFB  is  equal  to  the  angle 
AEF,  FB  cannot  meet  AB,  nor  fall  below  FD,  and  conse- 
quently must  coincide  with  the  parallel  CD  (Def.  30):  and 
I  nee,  the  alternate  angles  AEF  and  EFD  are  equal. 


Cor.  U  a  line  be  perpendicular  to  one 
of  two  parallel  lines,  it  will  also  be  pcr- 
pendiculAT  to  the  other 


B  0  O  K     1.  27 


Of    Parallel    Lino 


THEOREM    XIII. 

Conversely, — If  a  line  intersect  two  straight  lines,  making  thi 
alternate  angles  equal,  those  straight  lines  mil  be  parallel. 

fict  tlic  line  LF  meet  the  lines  AB, 
CD.  making  the  angle  AEF  equal  to  the 
fto.^lo  EFD :  then  will  the  lines  AB  and 
CD  be  j)arallcl.  0     7/:^\~~7) 

For,  if  they  arc   not  parallel,  suppose  G 

through  the  point  F  the  line  FO  to  be  drawn  parallel  to  AB. 

Then,  because  of  the  parallels  AB,  FGy  the  alternate  angles, 
AEF  and  EFG  will  be  equal  (Th.  xii).  Rut,  by  nypoihesis, 
the  angle  AEF  is  equal  to  EFD:  hence,  the  angle  EFD  id 
equal  to  the  angle  EFG  (Ax.  1) ;  that  is,  a  part  is  equal  to  the 
whole,  which  is  absurd  (Ax.  8) :  therefore, no  line  but  CD  can 
be  parallel  to  AB. 

Cor.  If  two  lines  are  perpendicular  tc 
the  same  line,  they  will  be  parallel  to 
each  other. 


THEOREM    XIV. 

If  a  line  cut  two  parallel  lines,  the  outward  angle  is  equal  to 
the  inward  opposite  angle  on  the  same  .side;  aid  the  two  inward 
angles,  on  the  same  side,  are  equal  to  two  right  angles. 

Let  tlie  line  EF  cut  the  two  parallels 
A  B     CD  .  tlien  will  the   outward  angle  y 

EGB  be  cquai  to  the  inward  opposite  an-       A  ^/^^     ^ 

glo  EHD  ;  and  the  two   inward  angles,    C ~/^^^ 7 

nCH  and    GllD,    will  be  equal  to  two  ^^ 

Tight  angles. 


28  GEOMETRY 


Of   Parallel    Lines 


First.  Since  the  lines  AB,  CD,  are  parallel,  the  anglo  AGH 
IS    equal    to    the   alternate    angle    GHD  E 

(Til.   xii) ;  but  the   angle  AGH  is  equal   ^ y^' 

to  the  opposite  angle  EGB  :  hence,  the 


C     /H  D 

angle  EGB   is  equal  to  the  angle  EHD         y 

(Ax.  1). 

Secondly.  Since  the  two  adjacent  angles  EGB  and  BGli 
are  equal  to  two  right  angles  (Th.  ii) ;  and  since  tlie  angle 
EGB  has  been  proved  equal  to  EHD,  it  follows  that  the  sum 
of  BGH  plus  GHD,  is  also  equal  to  two  right  angles. 

Cor.  1.  Conversely,  if  one  straight  line  meets  two  other 
straight  lines,  making  the  angles  on  the  same  side  equal  to 
each  other,  those  lines  will  be  parallel. 

Cor.  2.  If  a  line  intersect  two  other  lines,  making  the  sum 
of  the  two  inward  angles  equal  to  two  right  angles,  those  two 
lines  will  be  parallel 

Cor.  3.  If  a  line  intersect  two  other  lines,  makmg  the  sum 
(jf  the  two  inward  angles  less  than  two  right  angles,  those 
hnes  will  not  be  parallel,  but  will  meet  if  sufficiently  produced. 

THEOREM   .\v. 

All  straight  lines  which  are  parallel  to  the  same  line,  are  parallel 
to  each  other. 

Let  the  lines  AB  and  CD  be  each  par- 
allel to  EF:  then  will  they  be  parallel 
to  each  other. 

For.  let  the  line  GI  be  drawn  perpen- 
dicular to  EF :  then  will  it  also  be  per- 
pendicular to  the  parallels  AB  CD  (Th. 
fii  Cor.V 


< 

7 

A 

Ii 

C 

D 

E          I 

f' 

B  O  0  K     c .  29 


Of  Trianglca. 


Then,  since  the  lines  AB  and  CD  are  porpiiiJiciilar  to  the 
line  Gly  ihcy  will  be  parallel  to  each  other  (Th.  xiii.  Cor). 

THEOREM  XVI. 

If  one  side  of  a  tnangle  be  produced,  the  outward  angle  will  be 
C(]ual  to  the  sum  of  the  inward  opposite  angles. 

In  the  triangle  yl 5 C,  let  the  side   AB 
be  produced  to  D  :  then  will  the  outward               /^v  ^ 

angle  CBD  be  equal  to  the  sum  of  the  in-      y^      \ 
ward  opposite  angles  A  and  C.  ^ ^ jy 

For,  conceive  the  line  BE  to  be  drawn 
parallel  to  the  side  AC.     Then,  since  BC  meets  the  two  j)a- 
rallels  A  C,  BE,  the  alternate  angles  A  CB  and   CBE  will  be 
equal  (Th.  xii). 

And  since  the  line  AD  cuts  the  two  parallels  BE  and  AC 
the  angles  EBD  and  CAB  are  equal  to  each  other  (Th.  xiv) 
Therefore,  the  inward  angles  C  and  A,  of  the  triangle  ABC 
are  equal  to  the  angles  CBE  and  EBD ;  and  consequently 
the  sum  of  the  two  angles,  A  and  C,  is  equal  to  the  outward 
angle  CBD  (Ax.  1). 

THEOREM    XVII. 

In  any  tnangle  the  sum  of  the  three  anglts  is  equal  to  two  righ 

angles. 

Let  ABC  be  any  triangle:  then   will 
t;ic  sum  of  the  three  angles  C 

^  4- J5-hC= two  right  angles.  ^/    \ 

For,  let  the  side  AB  be  produced  \o  D       A  j] 

Then,  the  outward  angle 

CBD  Z.A+C  {Th.  xvi). 
3* 


30 


GEOMETRY 


Of   Triangles. 


To  each  of  these  equals  add  the  angle 
CBA,  and  we  shall  have 

CBD^  CBA=A+C-\-B. 
But  the  sum  of  the  two  angles   CBD 
and    CBA,  's  equal  to  two  right  angles  -^ 
(Th.ii):  hence 

A  \-B-^C=U\'o  right  angles  (Ax.  1). 

Cor.  1.  If  two  angles  of  one  triangle  be  equal  to  two  angles 
of  another  triangle,  the  third  angles  will  also  be  equal  (Ax.  3). 

Cor.  2.  If  one  angle  of  one  triangle  be  equal  to  one  angle 
of  another  triangle,  the  sum  of  the  two  remaining  angles  in 
each  triangle,  will  also  be  equal  (Ax.  3). 

Cor.  3.  If  one  angle  of  a  triangle  be  a  right  angle,  the  sum 
of  the  other  two  angles  will  be  equal  to  a  right  angle ;  and 
each  angle  singly,  will  be  acute. 

Cor.  4.  No  triangle  can  have  more  than  one  right  angle,  nor 
more  than  one  obtuse  angle ;  otherwise,  the  sum  of  the  three 
angles  would  exceed  two  right  angles ;  hence,  at  least  two 
angles  of  every  triangle  must  be  acute. 

THEOREM    XVIII. 

I.  A  perpendicular  is  the  shortest  line  that  can  be  drawn  from 
a  given  point  to  a  given  line. 

II.  If  any  number  of  lines  be  drawn  from  the  same  point,  thost 
which  arc  nearest  the  perpendicular  are  less  than  those  which  are 
more  remote. 

Let  ^  be  a  given  point,  and  BE  a 
straight  line.  Suppose  AB  to  be  drawn 
perpendiculai  to  DE,  and  suppose  the 
-fjlique    lines    AC    and    AD  also  to   be 


D  O  O  K     I 


3i 


Of    Triangles 


drawn:  Then,  AD  will  be  shorter  than  ciihcr  of  the  oblique 
lines,  and  AC  will  be  less  than  AD 

First.  Since  the  angle  Z?,  in  the  triangle  .4  CB,  is  a  righ 
angle,  the  angle  C  will  be  acute  (Th.  xvii.  Cor.  3) :  and  since 
llie  greater  side  of  every  triangle  is  opposite  the  greater  angle 
(Th.  xi),  the  side  AC  will   be  greater  than  AB. 

Stcondbj.  Since  the  angle  ACB  is  acute,  the  adjacent  angle 
ACD  will  be  obtuse  (Th.  ii) :  consequently,  the  angle  2)  is 
acute  (Th.  xvii.  Cor.  3),  and  therefore  less  than  the  an^lo 
ACD.  And  since  the  greater  side  of  every  triangle  is  oppo- 
site the  greater  angle,  it  follows  that  AD  is  greater  than  AC. 

Cor.  A  perpendicular  is  the  shortest  distance  from  a  point 
to  a  line. 


THEOREM    XIX. 

//  two  right  angled  triangles  have  the  hypothenuse  and  a  sid^ 
of  the  one  equal  to  the  hypnlhemtse  and  a  side  of  the  other,  the 
reirMtning  parts  will  also  he  rr/ual,  each  to  each. 

I>ct  the  two  rijzlit  auiilcd  trianoles 

or?  '^ 

ABC  and  DEI'\  have  the  hypothe- 
nuso  AC  equal  to  DF,  and  the  side 
A  B  equal  to  DE :  then  will  the  re- 
maining parts  be  equal,  each  to  each.  ^ 

For,  if  the  sido  BC  is  equal  to  EF,  the  correspond! rig  an- 
gles of  the  two  triangles  will  be  equal  (Th.  viii).  If  the  sides 
Ere  unequal,  suppose  BC  to  be  the  greater,  and  take  a  part, 
EG  equal  to  EF,  and  draw  AG. 

Then,  in  the  two  triangles  ABG  and  DEF  the  angle  B  ig 
equal  to  the  angle  E,  the  side  AB  io  the  side  DE,  and  the  side 
f^G  to  the  sido  EF:  hence,  the  two  triangles  are  equal  in  al! 
respects  (Th.  iv)  and  consequently,  the  side  ^G  is  e<iuai  to 


G  a 


32  GEOMETRY. 


Of    Polygons, 


DF.  Bui  DF  is  equal  to  AC^  by  hypothesis;  therefore 
^6^  is  equal  to  AC  (Ax.  1).  But  this  is  impossible  (Tk 
irviii) ;  hence,  the  sides  UC  and  FF  cannot  be  unequal ;  con- 
sequently, the  triangles   are  equal  (Th.  viii). 

THEOREM    XX 

The  sum  of  tlie  four  angles  of  every  quadrilateral  is  equal  to  four 

right  angles. 

Let  A  CBD  be  a  quadrilateral :  then  will  p 

yl-4-J5+C-fZ)=: four  right  angles.  /^\. 

Let  the  diagonal  DC  be  drawn  dividing       4/     !        y^ 
iho  quadrilateral  AB,  into  two  triangles,  ^\y^^^^ 

BDC,  ADC.  C 

Then,  because  the  sum  of  the  three  angles  of  each  triangle 
is  equal  to  two  right  angles  (Th.  xvii),  it  follows  tliat  the  sum 
of  the  angles  of  both  triangles  is  equal  to  four  right  angles. 
But  the  sum  of  the  angles  of  both  triangles,  make  up  the  angles 
of  the  quadrilateral.  Hence,  the  sum  of  the  four  angles  of  the 
(quadrilateral  is  equal  to  four  right  angles 

Cor.  L  If  then  three  of  the  angles  be  right  angles,  the 
fourth  angle  will  also  be  a  right  angle 

Cor.  2.  If  the  sum  of  two  of  the  fom  ngles  be  equal  to  two 
right  angles,  the  sum  of  the  remaining  two  will  also  be  equal 
»o  two  right  angles. 

Cor.  3.  Since  all  the  angles  of  a  square  or  rectangle,  are 
equal  to  each  other  (Def.  48),  and  thei.  sum  equal  to  foui 
right  angles,  it  follows  that  each  angle  is  equal  to  one  right 
angle. 

THEOREM    XXI. 

The  sum  of  all  the  interior  angles  of  any  polygon  is  equal  to 
tmce  as  many  right  angles,  wanting  four,  as  the  figure  has 
side 


BOOK     I  . 


33 


Of    Polygons 


Let  ABODE  be  any  polygon:  then  will 
the  sum  of  its  inward  angles 

A-\-B+C-i-D-\-E  B 

be  equil  to  twice  as  many  right  angles, 
wanting  four,  as  the  figure  has  sides. 

For,  from  anj  point  P,  within  the  poly-        A  B 

gon,  draw  the  lines  PA,  PB,  PC,  PD,  ?E,  to  each  cf  the 
angles,  dividing  the  polygon  into  as  many  triangles  vjs  the 
figure  has  sides. 

Now,  the  sum  of  the  three  angles  of  each  of  these  triangles 
is  equal  to  two  right  angles  (Th.  xvii) :  hence,  the  sum  of  the 
angles  of  all  the  triangles  is  equal  to  twice  as  many  right  an- 
gles as  the  figure  has  sides. 

But  the  sum  of  all  thie  angles  about  the  point  P  is  equal  to 
four  right  angles  (Th.  ii.  Cor.  4) ;  and  since  this  sum  makes 
no  part  of  the  inward  angles  of  the  polygon,  it  must  be  sub- 
tracted from  the  sum  of  all  the  angles  of  the  triangles,  before 
found.  Hence,  the  sum  of  the  interior  angles  of  the  polygon 
is  equal  to  twice  as  many  right  angleSt  wanti?ig  four^  as  the  figure 
has  sides. 


Sch.  This  proposition  is  not  applicable 
to  polygons  which  have  re-entrant  angles. 

The  reasoning  is  limited  to  polygons 
with  salient  angles,  which  may  properly 
be  named  convex  polygons. 


THEOREM    XXII. 


If  every  side  of  a  polygon  he  produced  out,  the  sum  of  all  the  tna 
ward  angles  thereby  formed ,  mil  be  equal  to  fmr  righ:  angfrs 


34  G  E  O  M  E  r  il  i- 


Of    Polygons 


Let  A,  B,  C,  D,  and  E,  be  the  outward 
angles  of  a  polygon  formed  by  producing 
all  tlie  sides.     Then  will 

>i -h  5 +C-fZ)+£z=  four  right  angles. 

For,  each  interior  angle,  plus  its  exte- 
rior angle,  as  A-{-a,  is  equal  to  two  right 
ingles  (Th.  ii).     But  there  are  as  many  exterior  as  interioi 
angles,  and  as  many  of  each  as  there  are  sides  of  the  polygon : 
hence,  the  sum  of  all  the  interior  and  exterior  angles  will  be 
equal  to  twice  as  many  right  angles  as  the  polygon  has  sides. 

But  the  sum  of  all  the  interior  angles  together  with  four  right 
angles,  is  equal  to  twice  as  many  right  angles  as  the  polygon 
has  sides  (Th.  xxi) :  thcit  is,  equal  to  the  sum  of  all  the  in- 
ward and  outward  angles  taken  together. 

From  each  of  these  equal  sums  take  away  the  inward  angles, 
and  there  will  remain,  the  outward  angles  equal  to  four  right 
angles  (Ax.  3). 

THEOREM    XXIII 

The  opposite  sides  and  angles  of  every  paraRclogram  are  equals 
each  to  each :  and  a  diagonal  divides  the  parallelogram  into  two 
equal  triangles. 

Let  ABCD  be  any  parallelogram,  and 
DB  a  diagonal:  then  will  the  opposite 
sides  and  angles  be  equal  to  each  other, 
each  to  each,  and  the  diagonal  DB  will 
divide  the  parallelogram  into  tw^o  equal 
triangles. 

For,  since  the  figure  is  a  parallelogram,  the  sides  AB^  DC 
are  oarallel.  as   also  the  sides    AD,    BC.     Nonv,  since   thf 


R  n  o  K    1 .  3o 


Of   Parallelograms. 


parallels  arc  cut  by  the  diagonal  DBy  the  alternate  angles  will 
be  equal  (Th.  xii) :  that  is  the  angle 

ADB-DBC         and         BDC-ABD. 

Hence  the  two  triangles  ADB  i5Z)C,  having  two  angles  in 
the  one  equal  to  two  angles  in  the  other,  will  have  their  third 
angles  ccjual  (Th.  xvii.  Cor.  1),  viz.  the  angle  A  equal  to  the 
angle  C,  and  these  are  two  of  the  opposite  angles  of  the 
parallelogram. 

Also,  if  to  the  equal  angles  ADB,  DBC,  we  add  the  equals 
BDC,  ABD,  the  sums  will  be  equal  (Ax.  2) :  viz.  the  whole 
angle  ADC  to  the  whole  angle  ABC,  and  these  are  the  other 
two  opposite  angles  of  the  parallelogram. 

Again,  since  the  two  triangles  ADB,  DBC,  have  the  side 
DB  common,  and  the  two  adjacent  angles  in  the  one  eijual  to 
the  two  adjacent  angles  in  the  other,  each  to  each,  the  two 
triangles  will  be  equal  (Th.  v) :  hence,  the  diagonal  divides 
the  parallelogram  into  two  equal  triangles. 

Cor.  1.  If  one  angle  of  a  parallelogram  be  a  right  anglo, 
each  of  the  angles  will  also  be  a  right  angle,  and  the  parallelo- 
gram will  be  a  rectangle. 

Cor.  2.  Hence,  also,  the  sum  of  either  two  adjacent  angles 
of  a  parallelogram,  will  bo  equal  to  two  right  angles. 


THEOREM     XXIV. 

If  the  opposite  sides  of  a  quadrilateral,  are  equal,  each  to  each^ 
the  €fjual  sides  wUl^e  parallel,  and  the  figure  will  be  a  |» 
rallelogram. 


m 


GEOMETRY 


Ol    Parallelograms 


Lei  ABCD  be  a  quadrilateral,   having 
its  opposite  sides  respectively  equal,  viz. 
AB=CD         and         AD  =  BC 

then  will  these  sides  be  parallel,  and  the    ^ 
figure  will  be  a  parallelogram. 

For,  draw  the  diagonal  BD.  Then,  the  two  triangles  A  81) 
BDC,  have  all  the  sides  of  the  one  equal  to  all  the  sides  of 
the  other,  each  to  each :  therefore,  the  two  triangles  are  equal 
(Th.  viii) ;  hence,  the  angle  ADB,  opposite  the  side  AB,  ia 
equal  to  the  angle  DBC  opposite  the  side  DC ;  therefore,  the 
sides  AD,  BC,  are  parallel  (Th.  xiii).  For  a  like  reason  DC 
is  parallel  to  AB,  and  the  figure  ABCD  is  a  parallelogram. 


THEOREM    XXV. 
If  two  opposite  sides  of  a  quadrilateral  are  equal  and  parallel ^ 
*he  remaining  sides  will  also  he  equal  and  parallel,  and  the  figure 
will  be  a  parallelogram. 

Let  ABCD  be  a  quadrilateral,  having 
the  sides  AB,  CD,  equal  and  parallel: 
then  will  tlie  figure  be  a  parallelogram. 

For,  draw   the   diagonal  DB,  dividing 
the  quadrilateral  into  two  triangles.    Then, 
GJnce  ^S  is  parallel  to  DC,   the  alternate   angles,  ABD  and 
BDC  are  equal  (Th.  xii) :  moreover,  the  side  BD  is  common ; 
hence  the  two  triangles  have  two  sides  and  the  included  ang.t 
of  the  one,  equal  to  two  sides  and  the  included  angle  of  the 
Other:  the  triangles  are  therefore  equal,   and   consequently 
AD  is  equal  to  BC,  and  the  angle  ADB  to  the  angle  DBC 
and  consequently,   AD  is    also   parallel  io  BC  (Th     xiii 
Tlierei'bre,  the  figure  ABCD  is  a  parallelogram. 


B  0  O  K     1  37 


Of    Parallelograms 


THEOREM    XXVI. 

TJu  two  diagonals  of  a  parallelogram  divide  each  other  into  tqua. 
parts,  or  mutually  bisect  each  otlter. 

Lc*  A  BCD  be  a  parallelogram,  and 
ACf  Bl)  Its  two  diagonals  intersecting  at 
E.     Then  will 

AE  =  EC         and         BE=ED. 

A 

Comparing  the  two  triangles  AED  and 
BEC,  we  find    the   side    AD— EC    (Th.    xxiii),  the   angle 
ADE  =  EBC  and  EAD=ECB :  hence,  the  two  triangles  are 
equal  (Th.  v) :  therefore,   AE^   the  side  opposite  ADE,  is 
equal  to  EC,  the  side  opposite  EBC;  and  ED  is  equal  to  EB 

Sch.  In  the  case  of  a  rhombus  (Def.  48), 
the  sides  AB,  BC  being  equal,  the  trian- 
gles AEB  and  BEC  have  all  the  sides  of 
the  one  equal  to  the  corresponding  sides 
of  the  other,  and  are  therefore  equal. ^ 
Whence  it  follows  that  the  angles  AEB 
and  BEC  are  equal.  Therefore,  the  diagonals  of  a  rhombu 
biBOct  each  other  at  right  angles. 


G  E  O  M  E  T  K  y  . 


BOOK   II, 


OF      THE       CIRCLE 


DEFINITIONS. 


1.  The  ciicumferencc  of  a  circle  is  a  curve  line,  all  the 
points  of  whicli  are  equally  distant  from  a  certain  point  within 
called  the  centre. 

2.  The  circle  is  the  space  bounded  by  this  curve  line. 

3.  Everystraightline,CA,CZ>,C£,  drawn        ^ K^ 

from  the  centre  to  the  circumference,  is 
called   a   radius  or  semidia7neter.     Every      / 
line  which,  like   AB^  passes  through  the 
centre   and    terminates  in   the   circumfe- 
rence, is  called  a  dia?neter. 

4.  Any  portion  of  the  circumference, 
as  EFG,  is  called  an  arc. 

5.  A  straight  line,  as  EG,  joining  tho-^ 
extremities  of  an  arc,  is  called  a  chord. 

6  A  segment  is  the  surface  or  portion 
of  a  circle  included  between  an  arc  and 
its  chord.     Thus    EFG  is  a  segment. 


BOOK     II. 


39 


D  e  fi  n  i  t  i  on  s 


7.  A  sector  is  liie  part  of  the  circle  in- 
cliidod  between  an  arc  and  the  two  radii 
drawn  through  its  extremities.  Thus, 
CA  5  is  a  sector 


8.  A  straight  line  is  said  to  be  in-      m 
scribed  in  a  circle,  A\iien  its  extremities       ^ 
are  in  the  circumference.     Thus,  the 
line  AB  is  inscribed  in  a  circle. 


9.  An  inscribed  angle  is  one  which 
is  formed  by  two  chords  that  intersect 
each  other  in  the  circumference.  Thus, 
BAC  is  an  inscribed  angle. 


10.  An  inscribed  triangle  is  one 
which  has  its  three  angular  points  in 
the  circmiiference.  Thus,  ABC  is  an 
inscribed  triangle.  U 


11.  Any  polygon  is  said  to  be  in- 
scribed in  a  circle  when  the  vertices  of 
all  the  angles  are  in  the  circumference. 
The  ciicle  is  then  said  to  circumscribe 
iho  polygon. 


40 


GEOMETRY 


Definitions 


12  A  secant  is  a  line  which  meets  the 
circumference  in  two  points,  and  lies 
^nartly  within  and  partly  without  the 
circle.     Thus   ^  S  is  a  secant. 


iM 


13.  A  tangent  is  a  line  which  has 
but  one  point  in  conunon  with  the  cir- 
cumference.    Thus,  CMB  is  a  tangent.       \^ 


14.  Two  circles  are  said  to  touch 
each  other  internally,  when  one  lies 
within  the  other,  and  their  circumfe- 
rences have  but  one  point  in  common. 


15.  Two  circles  are  said  to  touch 
each  other  externally,  when  one  lies 
without  the  other,  and  their  circumfe- 
lences  have  but  one  point  in  common 


BOOK     II 


41 


Of    the    Circle. 


THEOREM    I. 

A  diameter  is  greater  than  any  other  chord. 

Let  AD  l>c  any  chord.  Draw 
the  radii  CA,  CD  to  its  extremities. 
We  shall  then  have  A  G-\-  CD  greater 
than  AD  (Book  I.  Th.  X*).  But 
AC-\-CD  is  equal  to  the  diameter 
AB :  hence,  tlie  diameter  AB  is 
greater  than  AD, 


THEOREM    II. 

If  from  tkc  centre  (f  a  circle  a  line  be  drawn  to  the  middle  oj 
a  chord, 

I .  It  mil  be  perpendicular  to  the  chord ; 

II.  And  it  will  bisect  the  arc  of  the  chord. 
Let  C  be  the  centre  of  a  circle,  and 

AB  any  chord.  Draw  CD  through 
D,  the  middle  point  of  the  chord,  and 
produce  it  to  E:  then  will  CD  be 
perpendicular  to  the  chord,  and  the 
arc  AE  equal  to  EB. 

First.  Draw  the  two  radii  CA,  CB. 
Then  the  two  triangles  ACD^  DCB, 
have  the  three  side  s  of  the  one  equal  to  the  tliree  sides  of  the 


*NoU.  NMien  reference  if  made  fronr.  one  theorem  to  another,  in  the 
same  Book,  the  number  of  ihe  thoorem  refertftd  to  is  aloue  c^iven  •  but 
when  the  theorem  rcferrol  to  is  found  ir  a  preceding  Book,  the  number  of 
the  Book  is  also  w'ven. 


42 


G  E  O  31  E  T  R  y  . 


Of   th 


le. 


other,  each  to  each:  viz.  AC  equal  to 
CB,  being  radii,  AD  equal  to  DB,  by 
hypothesis,  and  CD  common:  hence, 
tlie  corresponding  angles  are  equal 
(I Jock  I.  Th,  viii) :  that  is,  the  angle 
CDA  equal  to  CDB,  and  the  angle 
ACD  equal  to  the  angle  DCB. 

But,  since  the  angle   CDA  is  equal 
to  the  angle   CDB,  the  radius  CE  is   perpendicular  to   tbe 
chord  AB  (Bk.  1.  DeL  20). 

Secondly.  Since  the  angle  ACE  is  equal  to  5 Ci?,  the 
arc  A  E  will  be  equal  to  the  arc  EB,  for  equal  angles  must 
have  equal  measures  (Bk.  I.  Def.  29). 

Hence,  the  radius  drawn  through  the  middle  point  of  a  chord, 
is  perpendicular  to  the  chord,  and  bisects  the  arc  of  the  chord. 

Cor.  Hence,  a  line  which  bisects  a  chord  at  right  angles, 
bisects  the  arc  of  the  chord,  and  passes  through  the  centre  o( 
fhe  circle.  Also,  a  line  drawn  through  the  centre  of  the  cir- 
cle and  perpendicular  to  the  chord, bisects  it. 

tiieore:\i   III. 

If  more  than  tiro  equal  lines  can  he  drawn  from  any  point  witfan 
a  circle  to  the  circiunfejence^  that  point  will  be  the  centre. 

Let  D  be  any  point  within  the  circle 
ABC.  Then,  if  the  three  lines  DA, 
DB,  and  DC,  drawn  from  the  point  D 
to  the  circumference,  are  equal,  the 
point  D  will  be  the  centre. 

For,  draw  the  chords  AB,  BC,  bi- 
sect them  at  the  points  E  and  F,  and 
ioin  DE  and  DF. 


BOOK      11 


43 


Of   the    Circle 


Then,  since  llic  two  triangles  DAE  and  DEB  have  the  side 
AE  equal  to  EB,  AD  equal  to  DB,  and  DE  coi^mon,  ihey 
wiU  be  equal  in  all  respects ;  and  consequently,  the  angle 
DEA  is  equal  to  the  angle  DEB  (Bk.  1.  Th.  viii) ;  and 
therefore,  DE  is  perpendicular  to  AB  (Bk.  I.  Dei  20)  But 
if  DE  bisects  AB  at  right  angles,  it  wih  pass  through  the 
centre  of  the  circle  (Th.  ii.  Cor). 

In  like  manner,  it  may  be  shown  that  DF  passes  through 
the  centre  of  the  circle,  and  since  the  centre  is  found  in  the 
two  lines  EDy  DF,  it  will  be  found  at  tlieir  common  inter- 
section D. 


THEOREM     IV. 

Ant/  chords  which  are  equally  distant  from  ike  centre  of  a  arcltt, 
are  equal. 

i^ei  AB  and  ED  be  two  chords  equally 
distant  from  the  centre  C:  then  will  the 
two  chords  AB,  ED  he  equal  to  each 
other 

Draw  CF  perpendicular  to  AB,  and 
CG  perpendicular  to  ED,  and  since  these 
perpendiculars  measure  the  distances  from 
the  centre,  they  will  be  equal.  Also  draw 
CB  and  CE. 

Then,  the  two  right  angled  triangles  CFB  and  CEG  hav 
ing  the  hypothenuse  CB  equal  to  the  hypothenuse  CE,  anH 
the  side  CF  equal  to  CG,  will  have  the  third  side  BF  equal  tc 
EG  (Bk.  I  Th.  xix)  But,  BF  is  the  half  of  BA  and  EG 
the  half  cf  DE  (Th.  ii.  Cor);  hence  BA  is  equal  to  DE 
(Ax   6). 


44 


GKORIET  RY. 


Of    the    Circl 


THEOREM    V. 

A   line  which   is  perpendicular  to  a  radius  at  its  extremity^  is 

tangent  to  the  circle. 

Let  the  line  ABD  be  perpendicular 
to  the  radius  CB  at  the  extremity  B  : 
then  will  it  be  tangent  to  the  circle  at 
the  point  B. 

For,  from  any  other  point  of  the 
line,  as  D,  draw  DFC  to  the  centre, 
cutting  the  circumference  in  F. 

Then,  because  the  angle  B,  of  the 
triangle  CDB,  is  a  right  angle,  the  angle  at  D  is  acute  (Bk  1. 
Th.  xvii.  Cor.  3),  and  consequently  less  than  the  angle  B. 
But  the  greater  side  of  every  triangle  is  opposite  to  the  greatei 
angle  (Bk.  I.  Th.  xi) ;  therefore,  the  side  CD  is  greater  than 
CB,  or  its  equal  CF.  Hence,  the  point  D  is  without  the  cir- 
cle, and  the  same  may  be  shown  for  every  other  point  of  the 
line  AD.  Consequently,  the  line  ABD  has  but  one  point  in 
common  with  the  circumference  of  the  circle,  and  therefore 
IS  tangent  to  it  at  the  point  B  (Def.  13) 

Cor.  Hence,  if  a  line  is  tangent  to  a  circle,  and  a  radius  be 
drawn  through  the  point  of  contact,  the  radius  will  be  perpen 
dicular  to  the  tangent. 


THEOREM    VI. 

if  the  distance  between  the  centies  of  two  circles  is  equal  to 
the  sum  of  their  radii,  the  two  circles  will  touch  each  other 
externally. 


BOOK      1 r 


45 


Of    the    Circle 


^ 


Let  C  and  D  be  the  two  centres,  and 
suppose  the  distance  between  them  to 
be  equal  ♦.o  the  sum  of  tlie  radii,  that  is, 
to  CA-\  AD 

The  circumferences  of  the  circles 
will  ey  idently  have  the  point  A  common,  and  they  will  have  n«j 
oilier.  Because,  if  they  had  two  points  common,  that,  is  if  ihey 
cut  each  other  in  two  points,  G  and  H,  the  distance  CD  be- 
tween their  centres  would  be  less  than  the  sum  of  their  radii 
CH,  HD  (Bk.  I.  Th.  x) ;  but  this  would  be  contrary  to  the 
supposition. 


THEOREM    VII. 

//  the  distance  between  the  centres  of  two  circles  is  equal  to 
the  difference  of  their  radii,  the  two  circles  will  touch  each  otk'^ 
interna  II I/. 

Let  C  and  D  be  ilie  centres  of  two 
circles  at  a  distance  from  each  other 
equal  to  AD- A C=  CD.  Fr 

Now,  it  is  evident,  as  in  the  last  theo- 
rem, that  the  circumferences  will  have  the  ^  ^  ^  "'^ 
point  A  common  ;  and  they  can  have  no 
other.  For,  if  they  had  two  points  common,  the  difference  be- 
tween the  radii  AD  and  FC  would  not  be  equal  to  CD,  the 
distance  between  their  centres  :  therefore,  they  cannot  have 
two  poir.ts  in  common  when  the  difference  of  their  radii  id 
equal  to  the  distance  between  their  centres :  hence,  they  are 
tangent  to  each  other. 

Sch  If  two  circles  touch  each  other,  either  externally  oi 
internally,  their  centres  and  the  point  of  contact  will  be  in  thf 
same  straight  line 


46 


G  E  O  I\I  £  T  R  y 


Of   the    Circle 


THEOREM     VIII 


A  n  angle  at  the  circumference  of  a  circle  is  measured  by  half  i  it 
arc  that  subtends  it 


Let  BAD  be  an  inscribed  angle  :  llicn 
will  it  be  mccisiircd  by  half  the  arc  BED, 
which  subtends  it. 

For,  through  the  centre  C  draw  the 
diameter  ACE,  and  draw  the  radii  BCj 
CD. 

Then,  in  the  triangle  ABC,  the  exte- 
rior angle  BCE  is  equal  to  tlie  sum  of 
the  interior  angles  B  and  A  (Bk.  1.  Th.  xvi).     But  since  the 
triangle  BAC  is  isosceles,  the  angles   A   and  B  are  equal 
(Bk.  I.  Th.  vi) ;  therefore,  the  exterior  angle  BCE  is  equal 
to  double  the  angle  BAC. 

But,  the  angle  BCE  is  measured  by  the  arc  BE,  which 
euhtends  it ;  and  consequenily,  the  angle  BAE,  which  is  hali 
of  BCE,  is  measured  by  half  the  arc  BE. 

It  may  be  shown,  in  like  manner,  that  the  angle  EAD  is 
measured  by  half  the  arc  ED :  and  hence,  by  the  addition  of 
equals,  it  would  follow  that,  the  angle  BAD  is  measured  Ly 
half  the  arc  BED,  which  subtends  it. 


Cor.  1.  Hence,  if  an  angle  at  the  centre^  and  an  angle  nt  the 
circumference,  both  stand  on  the  same  arc,  the  angle  at  the 
ccntie  will  be  double  the  ansle  at  the  circumference. 


Cor.  2.  If  two  angles  at  the  circumference  stand  on  equaj 
arcH  thev  will  he  equal  to  each  other. 


BOOK     II 


47 


Of   the    Circle 


THEOREM    n. 

A  U  angles  at  the  arcumfcrcncc^  which  stand  upon  *.hf  same  art 
are  equal  to  each  other. 

F.et  the  angles  BA  C,BDC,  BFC,  liavo 
their  vertices  in  ihe  circumference,  and 
3tand  on  the  same  arc  BEC :  then  will 
they  be  equal  to  each  other. 

For,  each  angle  is  measured  by  lialf 
the  arc  BEC  (Th.  viii) ;  hence,  the  an- 
gles are  all  equal. 


THEOREM    X. 
An  angle  in  a  semicircle,  is  a  right  angle. 

Let  ABBC  be  a  semicircle  :  then  will 
every  angle,  as  B,  B,  inscribed  in  it,  be 
a  right  angle.  / 

For,    each  angle  is  measured  by  half  j 
the  semic'rcumference  yiZ)C,  that  is,  by  a 
quadiant,  which  measures  a  right  angle 
CBk    1.  Th.  i.  Cor.  2). 


THEOREM    XI. 

If  a  quadrilateral  he  inscribed  in  a  circle,  the  sum  of  either  two 

of  Its  opposite  angles  is  equal  to  two  right  angles. 

Let  A  BCD  be  any  quadrilateral  in- 
scribed in  a  circle ;  then  will  the  sum  of 
the  two  opposite  angles,  A  and  C,  or  B 
and  D,  be  equal  to  two  right  angles.  A  \J 

For,  the  angle  A  is  measured  by  half      ^^- — '  ^^ 

the  arc  DCB,  wliich  subtends  it  (Th.  viii) ; 


»'.^ 


Z?\ 


4S  a  E  0  M  E  T  R  y . 

Of    the    Circle. 

and  the  angle  C  is  measured  by  half  the 

arc    DAB,  which    subtends  it.     Hence, 

iho  sum   of  the  two  angles,  A  and  C.  is 

meaiiured  by  half  the  entire  circumference. 

Bat  half  the  entire  circumference   is  the 

measure  of  two  right  angles  ;  therefore, 

tho  sum  of  the  opposite  angles  A  and  C  is  equal  to  two  rigbl 

angles. 

In   like  manner,   it  may  be    shown,  that   the  sum  of  the 
wo  angles  B  and  D  is  equal  to  two  right  angles 


THEOREM    XII 

If  the  side  of  a  quadrilateral,  inscribed  in  a   circle,   be  pro- 
duced out,  the  exterior  angle  will  be  equal  to  the  inward  opposiit 

angle 

Let  the  side  BA,  of  the  quadrilateral 
aBCD  be  produced  to  E,  then  will  the 
outward  angle  DAE  be  equal  to  the  in- 


ward opposite  angle  C. 

E 

For,  the  angle  DAB  plus  the  angle  C, 

is  equal  to  two  right  angles  (Th.  xi).     But 
DA  B  plus  DAE  is  also  equal  to  two  right  angles  (Bk.  1.  Th.  ii). 
Taking  from  each  the  common  angle  DAB,  and  we  shall  have 
tlie  angle  DAE  equal  to  the  interior  opposite  angle  C. 


THEOREM    XIII. 

Two  parallel  chords  intercept  equal  arcs. 


BOOK     I  ! 


49 


Uf    the    Circle 


Let  the  chords  A  B  and  CD  be  parallel  : 
then  will  the  arcs  AC  and  BD  be  equal 

For^  draw  the  line  A  D.  Then,  because 
ihc  hnes  AB  and  CD  are  parallel,  the 
ailcinate  angles  ADC  and  DAB  will  be 
equal  (Bk.  I.  Th.  xii).  IJut  the  angle 
ADC  is  measured  by  half  the  arc  AC, 
and  the  angle  DAB  by  half  the  arc  BD  (Th.  viii) :  hence 
the  two  arcs  A  C  and  BD  are  themselves  equal. 


THEOREM   XIV. 

The  angle  formed  by  a  tangent  and  a  chords  is  measured  by  half 
the  arc  of  the  chord. 

Let  BAE  be  tangent  to  the  circle  at  the 
point  A,  and  AC  any  chord. 

From  At  the  point  of  contact,  draw  the 
diameter  AD. 

Then,  the  angle  BAD  will  be  a  right 
angle  (Th.  v.    Cor),  and  therefore  will  be 
measured  by  half  the  semicircle  AMD  iT 
(Bk.  I,  Th.  i.  Cor.  2). 

But  the  angle  DAC  being  at  the  circumference,  is  measure  1 
by  half  the  arc  DC:  hence,  by  the  addhion  of  equals,  the  two 
angles  BAD  and  DAC,  or  the  entire  angle  BAC  w'lU  be  moas- 
lucd  by  half  the  arc  AMDC. 

It  may  be  shown,  by  taking  the  difference  between  the  tAvo 
angles  DAE  and  DAC,  that  the  angle  CAE  is  measured  by 
lialf  the  arc  AC  included  between  its  sides. 
5 


60 


G  E  O  iM  E  T  il  Y  . 


Of    the    Circle. 


THEOREM    XV. 

If  a  tangent  and  a  chord  are  parallel  to  each  other,  they  will 
intercept  equal  arcs. 

Let  the  tangent  ABC  he  parallel  to  the 
chord  DF:  then  will  the  intercepted  arcs 
I)B,  BF,  be  equal  to  each  other. 

For,  draw  the  chord  DB.  Then,  since 
AC  and  DF  are  parallel,  the  angle  ABD 
will  be  equal  to  the  angle  BDF.  But 
ABD  being  formed  by  a  tangent  and  a 
chord,  will  be  measured  by  half  the  arc 
DB ;  and  BDF  beinor  an  ande  at  the  circumference  will  be 
measured  by  half  the  arc  BF  (Th.  viii).  But  since  the  angles 
are  equal,  the  arcs  will  be  equal :  hence  DB  is  equal  to  BF. 


THEOREM    XVI 

The  angle  formed  within   a  circle    by   the  intersection  of  two 
chords,  is  measured  by  half  the  sum  of  the  intercepted  arcs. 

Let  the  two  chords  AB  and  CD  inter- 
sect each  other  at  the  point  E :  then  will 
the  angle  AFC,  or  its  equal  DEB,  be 
measured  by  half  the  sum  of  the  inter- 
cepted  arcs  AC,  DB. 

For,  draw  the  chord  AF  parallel  to 
CD.  Then  because  of  the  parallels,  the 
f.ngle  DEB  will  be  equal  to  the  angle  FAB  (Bk  1.  Th.  xiv), 
and  the  arc  FD  to  the  arc  AC.  But  the  ang^le  FA  B  is  meas- 
ured by  half  the  arc  FDB,  that  is,  by  half  the  sum  of  lh»)  arcs 
FD,  DB.  Now,  since  FD  is  equal  to  ^C,it  follows  tliat  the 
angle  DEB,  ox  its  equal  AEC,  will  be  measured  by  lialf  tlu' 
Slim  of  the  arns  DB  and  A  G 


BOOK     II. 


Of  the    Circle. 


THEOREM    XVU. 

The  angle  formed  without  a  circle  by  the  intersection  cj 
two  secants  is  measured  by  half  the  difference  of  the  intcrcifted 
arcs. 

Let  the  two  secants  DE  and  EB  inter- 
sect eaeh  other  at  E :  then  will  the  angle 
DEB  be  measured  by  half  the  intercepted 
arcs  CA  and  DB. 

Draw  the   chord   AF  parallel  to  ED.  D/ 
Then,  because  AF  and  ED  arc  parallel, 
and  EB  cuts  them,  the  angles  FAB  and 
and  DEB  are  equal  (Bk.  I.  Th.  xiv). 

But  the  angle  FAB,  at  the  circumference,  is  measured  by 
half  the  arc  FB  (Th.  viii),  which  is  the  difference  of  the  arcs 
DFB  and  CA  :  hence,  the  equal  angle  E  is  also  measured  by 
half  the  difTerence  of  the  intercepted  arcs  DFB  and  CA 


THEOREM    XVIir. 

An    angle  formed  by    two    tangejits   is  measured   by  half  tliC 
difference  of  the  intercepted  arcs. 

Let  CD  and  DA  be  two  tangents  to 
the  circle  at  the  points  C  and  A  :  then 
will  the  angle  CDA  be  measured  by  half 
tlic  difference  of  the  intercepted  arcs  CEA 
and  CFA. 

For,  draw  the  chord  AF  parallel  to  the 
tangent    CD.      Then,  because  the  lines 
CD  -iud  AF  arc  parallel,  the  angle  BAF 
will  bo  equal  to  the  angle   BDC  (Bk.  I.  Th.  xiv).     But  the 
inerle  BAF,  formed  by  a  tanfjent  and  a  chord,  is  measured  by 


52 


G  E  O  iU  E  T  R  Y . 


Of   the    Circle 


half  the    arc   AF,  that   is,   by   half  the 
difference  of  CFA  and  CF. 

But  since  the  tangent  DC  and  the 
chord  AF  are  parallel,  the  arc  CF  is 
equal  to  the  arc  CA :  hence  the  angle 
BAF,  or  its  equal  BDC,  which  is  meas-/ 
ured  by  half  the  difference  of  CFA  and 
CF,  is  also  measured  by  half  the  differ- 
ence of  the  intercepted  arcs  CFA  and  CA. 


Ccr.  In  like  manner  it  may  be  proved 
that  the  angle  E,  formed  by  a  tangent  and 
secant,  is  measured  by  half  the  difference 
of  the  intercepted  arcs  AC  and  DBA. 


THEOREM     XIX 

The  chord  of  an  arc   of  sixty  degrees  is  equal  to  the  radius  of 
the  circle. 

"  et  AEB  be  an  arc  of  sixty  degrees 
and  AB  its  chord :  then  will  AB  be  equal 
to  the  radius  of  the  circle. 

For,  draw  the  radii  CB  and  CA. 
Then,  since  the  angle  ACB  is  at  the 
centre,  it  will  be  measured  by  the  arc 
AEB:  that  is,  it  will  be  equal  to  sixty 
degrees  (Bk.  I.  Def.  29). 

Again,  since  the  sum  of  the  three  angles  of  a  triangle  is 
equal  to  one  hundred  and  eighty  degrees  (Bk.  I.  Th.  xvii),  it 


BOOK     II 


63 


Of   the    Circle. 


follows  tliat  the  sum  of  the  two  angles  A  and  B  will  be  equil 
to  one  hundred  and  twenty  degrees.  But  the  triangle  CA  B 
is  isosceles:  hence,  the  angles  at  the  base  are  equal  (Bk.  I. 
I'h.  vi) :  hence,  each  angle  is  equal  to  sixty  degrees,  and 
consequently,  the  side  A  Bis  equal  io  AC  or  CB  (Bk.  I.  Th  vi). 


PROBLEIMS 


RELATING    TO    THE    FIRST    AND    SECOND    BOOKS. 


The  Problems  of  Geometry  explain  the  methods  of  con 
structmg  or  describing  the  geometrical  figures. 

For  these  constructions,  a  straight  ruler  and  the  common 
compasses  or  dividers,  are  all  the  instruments  that  are  ab- 
solutely necessary. 

DIVIDERS    OR    COMPASSES. 


The  dividers  consist  of  the  two  legs  5a,  Ac,  which  tum 
eftcily  about  a  common  joint  at  b.     The  legs  of  the  dividers 


•'!)4  G  E  O  31  E  T  R  Y 


Problems 


are  extended  or  brought  together  by  placing  the  forefinger  on 
the  joint  at  6,  and  pressing  the  thumb  and  fingers  a^ainf^t  the 
logs 

PROBLEM    1. 

On  any  line,  as  CD,  to  lay  off  a  distance  equal  to  A  B, 

Take  up  the  dividers  with  the 
fhumb  and  second  finger,  and  place 
the  forefinger  on  the  joint  at  6.  A  B 

Then,  set  one  foot  of  the  dividers       ^ 

at  A,  and  extend  the  legs  with   the        ' ~ ' ' 

thumb   and  fingers,  until  the   other 
foot  reaches  B. 

Then,  raise  the  dividers,  place  one  foot  at  C,  and  mark 
with  the  other  the  distance  CE :  and  tliis  distance  wiU  evi- 
dently be  equal  to  AB. 

TROBLEM    II. 

To  describe  from  a  given  centre  the  circumference  of  a  circle 
having  a  given  radius. 

Let  C  be  the  given  centre,  and 
CB  the  given  radius. 

Place  one  foot  of  the  dividers  at 
C  and  extend  the  other  leg  until  it 
reaches  to  B.  Tlien,  turn  the  di- 
viders around  the  leg  at  C,  and  the 
other  leg  will  descrilv?  the  required 
circumference 


BOOK     II 


55 


P  robl  cms. 


OF    THE     RULER. 


A  ruler  of  a  convenient  size,  is  about  twenty  inches  in 
length,  two  inches  wide,  and  one  fifth  of  an  inch  in  thickness. 
It  should  be  made  of  a  hard  material,  and  perfectly  straight 
and  smooth. 


PROBLEM    III. 

7  0  draw  a  straight  line  throvgh  two  given  potrJs  A  and  B. 

Place  one  edge  of  the  ruler  on 
A  and  slide  the  ruler  around  until 
he  same  edge  falls  on  B.     Then,  . 

with   a   pen,  or  pencil,  draw  the 
ine  AB. 


Jy' 


PROBLEM     IV. 

To  bisect  a  given  line:  that  is,  to  divide  it  uito  two  equal  parts, 

l,Gi  AB  be  the  given  line  to  be 
di\ided.  With  ^  as  a  centre,  and 
radius  greater  than  half  of  AB, 
describe  an  arc  IFE.  Then,  with 
2i  as  a  centre,  and  an  equal  radius 
BI  describe  the  arc  IHE.  Join 
the  points  /  and  E  by  the  line  IE. 
the  point  Z),  where  it  intersects 
AB,  will  be  the  middle  point  of  the 
Une  AB. 


56 


G  E  O  M  E  T  fl  Y  . 


Problems. 


For,  draw  the  radii  AI,  AE 
BI,  and  BE.  Then,  since  these 
radii  are  equal,  the  triangles  AIE 
•  nd  BJE  have  all  the  sides  of  the 
one  equal  to  the  corresponding  sides 
of  the  other  ;  hence,  their  corres 
ponding  angles  are  equal  (Bk  I. 
Th.  viii) ;  that  is,  the  angle  A  IE  is  equal  to  ihe  angle  BfE 
Therefore,  the  two  triangles  AID  and  BID,  have  the  sidi 
A  I— IB,  tlie  angle  AID  =  BID,  and  ID  common:  honcp 
thcv  are  equal  (Bk.  I.  Th.  iv),  and  AD  is  equal  to  DB. 

PROULExM    V. 

To  bisect  a  given  angle  or  a  given  ate. 

f^et  A  CB  be   the  given  angle, 
and  AEB  the  given  arc. 

From  the  points  A  and  B^  as 
centres,  describe  with  the  same 
radius  two  arcs  cutting  each  other 
in  D.  Through  D  and  the  centre 
C,  draw  CED,  and  it  will  divide 
the  angle  ACE  into  two  equal  parts,  and  also  bisect  the  arc 
AEB  at  E. 

'For,  draw  the  radii  AD  and  BD.     Then,  in  the  two  triangles 
ACD,  CBD,Me  have 

AC=CB,  AD  =  BD 

and  CD  common  :  hence,  the  two  triangles  have  their  corrrg. 
ponding  angles  equal  (Bk  I.  Th.  viii),  and  consequently,  A  CD 
is  equal  to  BCD.  But  since  ACD  is  equal  to  BCD,  it  fol 
lows  that  the  arc  AE,  which  measures  the  former,  is  equal  tc 
the  arc  BE.  which  measures  the  latter 


i)-. 


BOOK     II. 


67 


Problems. 


PROBLEM    VI. 
At  a  given  vomt  in  a  straight  line  tc  erect  a  perpendicular  to  tht 

line. 

Let  A  be  the  given  point,  and  BC 
the  given  line. 

From  A  lay  off  any  two  distances, 
AB  and  AC,  equal  to  each  other 
Then,  from  the  points  B  and  C,  as 
centres,  with  a  radius  greater  than 
AB,  describe  two  arcs  intersecting  each  other  at  D ;  draw 
DA,  and  it  will  be  the  perpendicular  required. 

For,  draw  the  equal  radii  BD,  DC.     Then,  the  two  trian- 
gles, BDA,  and  CD  A,  will  have 

AB=AC  BD=DC 

and  AD  common :  hence,  the  angle  DAB  is  equal  to  the  angle 
DAC  (Bk.  I.  Th.  viii),  and  consequently,  DA  is  perpendicu- 
lar to  5  C.  (Bk.  IDef.  21). 


SECOND    METHOD. 

When  the  point  A  is  near  the  extremity  of  the  line. 

Assume  any  centre,  as  P,  out  of 
the  given  line.  Then  with  P  as  a 
centre,  and  radius  from  P  to  ^,  de- 
scribe the  circumference  of  a  circle 
Through  C,  where  the  circumference 
cuts  BA ,  draw  CPD.  Then,  through 
D,  where  CP  produced  meets  the 
circumference,  draw  DA  :  then  will 

DA  be  perpendicular  to  BA,  since   CAD  is  an   angle  in   a 
dcmicirclo  (Bk.  11.  Th.  x). 


68 


GEOMETRY 


Problems 


PROBLEM     VII. 

Frrni  a  given  point  without  a  straight  line  to  let  fall  a  perpen 
dicular  on  the  line. 

Let  A  be  the  given  point,  and  BD 
the  given  line 

From  the  point  ^  as  a  centre,  with 
a  radius  greater  than  the  shortest 
distance  to  BD,  describe  an  arc  cut- 
ting BD  in  the  points  B  and  D. 
Then,vvith  B  and  D  as  centres,  and 
the  same  radius,  describe  two  arcs  intersecting  each  other  at 
E.     Draw  AFE,  and  it  will  be  the  perpendicular  required. 

For,  draw  the  equal  radii  AB,  AD,  BE  and  DE  Then, 
the  two  triangles  EAB  and  EAD  will  have  the  sides  of  the 
one  equal  to  the  sides  of  the  other,  each  to  each ;  hence,  their 
corresponding  angles  will  be  equal  (Bk.  I.  Th.  viii),  viz.  the 
angle  BAE  to  the  angle  DAE.  Hence,  the  two  triangles 
BAF  and  DAF  will  have  two  sides  and  the  included  angle  of 
the  one,  equal  to  two  sides  and  the  included  angle  of  the  other, 
and  therefore,  the  angle  AFB  will  be  equal  to  the  angle 
AFD  (Bk.  I.  Th.  iv) :  hence,  ^i^jG  will  be  perpendiculai 
lo  BD. 

SECOND    METHOD 

Wlien  the  given  point  A  is  nearly 
opposite  the  extremity  of  the  line. 

Draw  A  C,  to  any  point  C  of  the 
line  BD.  Bisect  ^C  at  P.  Then, 
with  P  as  a  centre  and  PC  as  a  ra- 
dius, describe  the  semicircle  CD  A  ; 
draw  ^D,  and  it  will  be  perpendicular 
to  CD  since  CD  A  is  an  angle  in  a  semicircle  (Bk.  II.  '1  h.  x). 


B  O  O  K    I  I  .  59 


Problems. 


PROBLEM    VI II. 

At  a  given  point  in  a  given  line,  to  make  an  angle  eq^ial  to  i 
given  angle 

Lot  A  bo  ilio  given  point,  AE 
Uio  given  line,  and  IKL  the  given 
angle. 

From  the  vertex  /{",  aa  a  ccnlre,    -^  ^ 

with  any  radius,  describe  llic  arc  /L,  terminating  in  the  two 
sides  of  the  angle  :  and  draw  the  chord  IL. 

From  the  point  ^4,  as  a  centre,  with  a  distance  AE,  equal 
to  KI,  describe  the  arc  DE :  then  with  £,  as  a  centre,  and  a 
radius  equal  to  the  chord  IL,  describe  an  arc  cutting  DE  at 
D;  draw  AD,  and  the  angle  EAD  will  be  equal  to  the 
angle  K. 

For,  draw  the  chord  DE.  Then  the  two  triangles  IKL 
and  EAD,  having  the  three  sides  of  the  one  equal  to  the  three 
sides  of  the  other,  each  to  each,  the  angle  EAD  will  be  equal 
to  the  angle  K  (Bk.  1.  Th.  viii). 

PROBLEM    IX. 

Through  a  given  point  to  draw  a  line  that  shall  be  parallel  to  a 
given  line. 

Let  A  be  the   given  point  and  /^ g 

RC  the  given  line. 

With  A  as  a  centre,  and  any  ra- 
dius greater  than  the  shortest  dis-  " 
lance  from  A  to  BC,  describe  the  indefinite  arc  DE.     From 
the  point  E,  as  a  centre,  with  the  same  radius,  describe  the 
arc  aF:  then,  make  ED  equa    to  AF  and  draw  AD,  and  it 
will  b^  the  required  parallel. 


60  GEOMETRY. 


B- 


Problems. 

For,  since  the  arcs  AF  and  ED 
are  equal,  the  angles  EAD  and 
AEFy  wliich  they  measure,  are 
equal :  hence,  the  line  AD  is 
parallel  to  BC  (Bk  1.  Th   xiii). 


PROBLEM    X. 

Two  angles  of  a  triangle  being  given  or  known ^  to  find  the.  thir^i 

Draw   the   indefinite    line 
DEF. 

At   any  point,  as    E,  make 
the  angle  DEC  equal  to  one  E 

of  the  given  angles,  and  then  CEH  equal  to  a  second,  by 
Prob.  VIII ;  then  will  the  angle  HEF  be  equal  to  the  third 
angle  of  the  triangle. 

For,  the  sum  of  the  three  angles  of  a  triangle  is  equal  to 
two  right  angles  (Bk.  I.  Th.  xvii) ;  and  the  sum  of  the  three 
angles  on  the  same  side  of  the  line  DE  i^  equal  to  two  right 
angles  (Bk.  I.  Th.  ii.  Cor.  2) :  hence,  if  DEC  and  CEH  are 
equal  to  two  of  the  angles,  the  angle  HEF  will  be  equal  to  the 
remaining  angle  of  the  triangle 

PROBLEM    XI. 

Three  sides  of  a  triangle  being  given,  to  describe  the  triangle 

Let  Aj  B,  and  C,  be  the  given 
fiides. 

Draw  DEj  and  make  it  equal  lo 
the  side  A.     From  the  point  D,  as 
a  centre,  with  a  radius  equal  ro  the        ^*" 
B*»cond   side  B,  describe  ar.  ajc  o 


B  O  O  K    1  1 .  6] 


Problems. 


from  £  as  a  centre,  with  the  third  side  C,  describe  another  arc 
intersecting  the  former  in  F:  draw  DF  and  FE:  then  will 
OFF  bo  the  required  triangle. 

For,  the  three  sides  are  respectively  equal  to  the  three  lines 
A.  B  and  C. 


PROBLEM     XII. 

TTie  adjacent  sides  of  a  parallelogram,  nnth  Che  angle  which  rh^iy 
contain,  being  given,  to  descrthe  the   narallelo^gram 

Let  ^  and  ^  be  the  given  sides  ,. 

and  C  the  given  angle.  / 


Draw  the  line  DFJ  and  make  it       i/. /£ 

equal  to  A.     At  the  point  B  make        ^'  '    /^ 

the  angle  FJBF  equa\  to  the  angle 

C.  Make  the  side  DF  equal  to  B.  Then  describe  two  arcs, 
one  from  i^  as  a  centre,  with  a  radius  FG  equal  to  DF^  tht 
other  from  F^  as  a  centre,  with  a  radius  FO  equal  to  DF. 
Through  the  point  G,  the  point  of  intersection,  draw  the  lines 
EG  and  FG,  and  DEGF  will  be  the  required  parallelogram. 

For,  in  the  quadrilateral  DFGE,  tlie  opposite  sides  DE 
and  FG  are  each  equal  to  A :  the  opposite  sides  DF  and 
EG  arc  each  equal  to  B,  and  the  angle  EDF  is  equa. 
to  C.  But,  since  the  opposite  sides  are  equal,  they  are 
also  parallel  (Bk.  1  Th.  xxiv),  and  therefore  the  figure  is  a 
arallclogram 


PROBLEM    XIII. 

To  describe  a  square  on  a  given  line. 
G 


r>2 


G  E  0  I\I  E  T  R  Y  . 


Problems. 


Let  AB  he  the  given  line. 

At  the  point  B  draw  5  C  perpendicu- 
lar to  AB,  by  Problem  VI,  and  then 
make  it  equal  to  AB. 

Tlien,  wiili  yl  as  a  centre,  and  ra- 
dius equal  to  AB,  describe  an  arc ;  and 
with  C  as  a  centre,  and  the  same 
radius  ^iB,  describe  another  arc;  and  through  D,  their  point 
of  intersection,  draw  AD  and  CD :  then  will  ABCD  be  the 
required  square. 

For,  smce  the  opposite  sides  are  equal,  the  figure  will  be  a 
parallelogram  (Bk.  I.  Th.  xxiv)':  and  since  one  of  the  angleo 
is  a  right  angle,  the  others  will  also  be  right  angles  (Bk.  1. 
Th.  xxiii.  Cor.  1) ;  and  since  the  sides  are  all  equal,  the  figure 
will  be  a  square. 


PROBLEM    XIV. 

To  construct  a  rhombus,  having  given   the  length  of  one  of  the 
equal  sides,  and  one  of  the  angles. 

Let  AB  be  equal  to  the  given  side, 
and  E  the  f^iven  an^le. 

At  B  lay  off  an  angle,  ABC,  equal 
to  E,  by  Prob.  VIIL  and  make  BC 
equal  to  AB.  Then,  with  A  and  C 
tis  centres,  and  a  radius  equal  io  AB,  ^ 
describe  two  arcs.  Through  D,  their  point  of  intersection, 
draw  the  lines  AD,  CD:  then  will  ABCD  be  the  required 
rhombus. 

For,  since  the  opposite  sides  are  equal,  they  will  be  par-'dlol 
iBk.  1.  Th.  xxiv).     But  they  are    each  equal  to  AB,  and  the 


BOOK    II 


«3 


Problems. 


angle  B  is  equal  to  the  angle  E :  hence,  ABCD  is  the  re- 
quired rhombus. 


PROBLEM    XV. 

To  find  the  centre  of  a  circle 

Draw  any  chord,  as  AB,  and  bisect  it 
by  Problem  IV.  Then,  through  F,  the 
middle  point,  draw  DCE,  perpendicular 
to  AB,  by  Problem  Vi.  Then  DCE 
will  be  a  diameter  of  the  circle  (Bk.  11. 
Th.  ii.  Cor.).  Then  bisect  DE  at  C, 
and  C  will  be  the  centre  of  the  circle. 


PROBLEM    XVI. 

Tn  describe   the    circumference  oj  a  circle  through   three  given 
points  not  in  the  same  straight  line. 

Let  A,  B,  C,  be  the  given  points. 

Join  these  points  by  the  straight 
lines  AC  AB,  BC. 

Then,  bisect  any  two  of  these 
straight  lines,  as  AB,  BC,  by  the 
perpendiculars  OD,  OP  (Prob.  iv) ; 
and  the  point  O,  where  these  per- 
pendiculars intersect  each  other, 
will  be  the  centre  of  the  circle. 

Then  with  O  as  a  centre,  and  a  radius  equal  to  OA,  de« 
scribe  the  circumference  of  a  circle,  and  it  will  jjass  through 
the  pohits  A,  B,  and  C. 

For,  the  two  right  angled  triangles  OA  P  and  OBP  have  the 
side  AP  equal  to  the  side  BP.  OP  common,  and  the  included 


64 


GEOMETRY. 


Pro  bl  ems. 


angles  OP  A  and  OPB  equal,  being 
right  angles ;  hence,  the  side  OB  is 
equal  to  OA  (Bk.  I.  Th.  iv). 

In  like  manner  it  may  be  shown 
that  0(7  is  equal  to  OB.  Hence,  a 
circumference  described  with  the 
radius  OA,  will  pass  through  the 
points  B  and  C. 

Sch.  This  problem  enables  us  to  describe  the  circumference 
of  a  circle  about  a  given  triangle.  For,  we  may  consider  the 
vertices  of  the  three  angles  as  the  three  points  through  which 
the  circumference  is  to  pass. 


PROBLEM    XVII. 

Through  a  given  point  in  the  circumference  of  a  circle,  to  drau 
a  tangent  line  to  the  circle. 

Let  A  be  the  given  point  j^ 

Through  A,  draw  the  radius  ^C  to  the 
centre,  and  then  draw  DAE  perpendicu- 
lar to  AC,  by  Problem  VI.  Then  will 
DAE  be  tangent  to  the  circle  at  the  point 
A  (Bk.  II.  Th.  ^) 


PROBLEM    XVIII. 


Thro*jgh   a  given  point   mthout  the  circumference,  to  draw  a 
tangent  hne  to  the  circle. 


BOOK     11. 


65 


Prob  1  ems 


Lcl  C  be  ihe  ceuirc  of  the  circle,  and 
A  ilie  given  point  williout  the  circle. 

Join  A  and  the  centre  C,  and  on  A  C 
as  a  il.ameter, describe  a  circumference. 
Through  the  points  B  and  D    where    / 
the  two  circuijiferences  intersect  each  / 
oilier,   draw   the   lines   AB  and   AD:  \ 
these  lines  will  be  tangent  to  the  circle 
»\'hose  centre  is  C. 

For,  since  the  angles  ABC  and 
ADC  are  each  inscribed  in  a  semicircle,  they  will  be  right 
angles  (Bk.  II.  Th.  x).  Again,  since  the  lines  AB,  AD, 
are  each  perpendicidar  to  a  radius  at  its  extremity,  they  will 
be  tanc^enl  to  the  circle  (Bk.  II.  Th.  v). 


PnOBLEM    XIX 

To  inscribe  a  circle  in  a  given  triano^lc. 

Let  ABC  be  the  given  tri- 
angle. 

Bisect  the  angles  A  and  B 
by  the  lines  AG  and  BO,  meet- 
ing at  the  point  O.  From  O, 
let  fall  the  perpendiculars  O/), 
0£,  OF,  on  the  three  sides  of 
the  triangle — these  perpendiculars  will  be  equal  to  each  other. 

For,  in  the  two  right  angled  triangles  DAO  and  FAO,  we 

ha/«  the  right  angle  D  equal  the  right  angle  F,  the  angle  FAO 

equal  to  DAO,  and  consequently,  the  third  angles  AOD  and 

AOF  are   equal  (Bk.   I.   Th    xvii.  Cor    1)       But   the  two 

triangle-s  havt,  a    common  side  AO,   hence,  they  are  eqnnl 

(Bk.   I.  Th    v),  and  consequently,  OD  is  equal  to  OF 
6* 


66 


GEOIMETRV. 


P  r  o  I)  1  e 


In  a  similar  manner,  it  may 
be  proved  that  OE  and  OD  are 
equal .  hence,  ilie  three  per- 
pendiculars, OD,  OF,  and  OE, 
are  all  eq\ial. 

•  Now,  if  with  0  as  a  centre,^ 
and  OF  as  a  radius,  we  describe 

the  circumference  of  a  circle,  it  will  pass  through  the  points 
D  and  E,  and  since  the  sides  of  the  triangle  are  perpendiculai 
to  the  radii  OF,  OD,  OE,  they  will  be  tangent  to  the  circum- 
ference (Rk.  II.  Th.  v).  Hence,  the  circle  will  be  inscribed 
in  the  triangle. 


PROBLEM    XX. 

To  inscribe  an  equilateral  triangle  m  a  circle. 

Through  the  centre  C  draw  any  diam- 
eter, as  ACB.  From  i?  as  a  centre,  with 
a  radius  equal  to  BC,  describe  the  arc 
DCE.  Then,  draw  AD,  AE,  and  DE, 
and  DAE  will  be  the  required  triangle. 

For,  since  the  chords  BD,  BE,  are 
(sach  equal  to  the  radius  CB,  the  arcs  BD,  BE,  are  each  equal 
to  sixty  degrees  (Bk.  II.  Th.  xix),  and  the  arc  DBE  to  one 
hundred  and  twenty  degrees:  hence,  the  angle  DAE  is  equal 
to   sixty  degrees  (Bk.  II.  Th.  viii). 

Again,  since  the  arc  BD  is  equal  to  sixty  degrees,  and  the 
tire  BDA  equal  to  one  hundred  and  eighty  degrees,  it  follows 
that  DA  will  be  equal  to  one  hundred  and  twenty  degrees : 
hence,  the  angle  DEA  is  equal  to  sixty  degrees,  and  consc* 
quently,   the   third  angle   ADE,   is  equal    to    sixty  degrees 


BOOK 


e*? 


Problems, 


Tlierefore,  tlie  triangle  ADE  is  equilateral  (Bk.  I.  Th.  vl 
Cor.  2). 


PROBLEM     XXI. 

To  inscribe  a  regular  hexagon  in  a  circle. 


Draw  any  radius,  as -AC.  Then  ap- 
ply the  radius  A  C  around  the  circum- 
ference, and  it  will  give  the  chords  AD, 
DE,  Eh\  FG,  GH,  and  HA,  which  m  ill 
be  the  sides  of  the  regular  hexagon.  For,  ^^      _^n 

A  l) 

the  side  of  a  hexagon  is  e(|iiai  to  ihe  radius  (iik.  II.  Th.  xix). 


PROBLEM     XXII. 

To  inscribe  a  square  tn  a  given  circle. 

Let  ABCD  be  the  given  circle. 
Draw  the  two  diameters  A  C,  BD,  at 
right  angles  to  each  other,  and  through 
the  points  A,  B,  C  and  D  draw  the 
lines  AB,  BC,  CD,  and  DA:  then 
will  ABCD  be  the  required  square. 

For,  the  four  right  angled  triangles, 
AOB,  BOC,  COD,  and  DOA  are 
equal,  since  the  sides  AO,  OB,  OC,  and  OD  are  equal,  bein|j 
radii  of  the  circle ;  and  the  angles  at  O  are  equal  in  each, 
being  right  angles :  hence,  the  sides  AB,  BC,  CD,  and  DA 
are  equal  (Bk.  I.  Th.  iv). 

But  each  of  the  angles  ABC,  BCD,  CD  A,  DAB,  is  a  right 
angle,  being  an  angle  in  a  semicircle  (Bk.  II.  Th  x) :  hence, 
the  figi'TP  ABCD  is  a  square  (Bk.  I.  Dcf   48) 


68 


G  E  0  I\l  E  T  R  Y 


Problems. 


Sch.  If  we  bisect  the  arcs  AB^ 
BCf  CD,  DA,  and  join  the  points, 
we  shall  liave  a  reorular  octagon  in- 
scribed  in  the  circle.  If  we  again 
bisect  the  arcs,  and  join  the  points  of 
bisection,  we  shall  have  a  regulai 
polygon  of  sixteen  sides. 


PROBLEM    XXIII. 


To  describe  a  square  about  a  given  circle. 


E 


H 


;  D  I 


Jl 


r>raw  the  diameters  AB,  DE,  at 
right  angles  to  each  other.  Through 
the  extremities  A  and  B  draw  FA  G 
and  HBI  parallel  to  DE,  and  through 
E  and  D,  draw  FEH  and  GDI  par- 
allel to  AB:  then  will  FGIHhe  the 
required  square. 

For,  since  ACDG  is  a  parallelogram,  the  opposite  sides  arc 
equal  (Bk.  I.  Th.  xxiii):  and  since  the  angle  at  C  is  a  right  angle 
all  the  other  angles  are  right  angles  (Bk.  I.  Th.  xxiii. Cor.  1): 
and  as  the  same  may  be  proved  of  each  of  the  figures  CI,  CH 
and  CF,  it  follows  that  all  the  angles,  F,  G,  I,  and  //,  are 
right  angles,  and  that  the  sides  GI,  IH,  HF,  and  FG,  are 
equal,  each  being  equal  to  the  diameter  of  the  circle.  Hence 
the  figure  GIHF  is  a  square  (Bk  I.  Def.  48). 


GEOMETRY. 


BOOK   III. 

OF      RjrriOB      AND      PROPORTIONS. 

DSP1N1TI0N8. 

1.  Ratio  is  the  quotient  arising  from  dividing  one  quantity 
by  another  quamity  of  the  same  kind.  Thus,  if  the  numbers 
3  and  6  have  the  same  unit,  the  ratio  of  3  to  G  will  be. 
expressed  by 

And  in  general,  if  A  and  B  represent  quantities  of  the  same 
kind,  the  ratio  of  yl  to  5  will  be  expressed  by 

B 

A 

2.  If  there  be  four  numbers,  2,  4,  0,  IG,  lijiviiig  such  values 
that  the  second  divided  by  the  first  is  equal  to  the  fourth  di- 
vided by  the  third,  the  numbers  are  said  to  be  in  proportion. 
A.nd  in  general,  if  there  be  four  quantities  -4,  B,  C,  and  D 
hftvijig  such  values  that 

B    D 
A=C' 

then,  A  is  said  to  have  the  same  ratio  to  By  that  C  has  to  D, 
or.  the  ratio  of  A  to  Z?  is  equal  to  the  ratio  of  C  to  D      When 


70  GEOMETRY 


Of   Uatios   and    ProportionB. 

four  quantities  have  this  relation  to  each  other,  they  are  said  to 
bo  in  proportion.  Hence,  the  proportion  of  four  quantities 
results  from  an  equality  of  their  ratios  taken  two  and  two . 

To  express  that  the  ratio  of  ^  to  5  is  equal  to  the   ratio 
of  C  to  D,  we  write  the  quantities  thus  : 
A     '.     B   ::    G  :  D  : 
and  read,  A  is  to  B,  as  G  to  D. 

The  quantities  which  are  compared  together  are  called  tne 
terms  of  the  proportion.  The  first  and  last  terms  are  called 
the  extremes,  and  the  second  and  third  terms,  the  means. 
Thus,  A  and  D  are  the  extremes,  and  B  and  G  the  means. 

3.  Of  four  proportional  quantities,  the  first  and  third  are 
called  the  antecedents,  and  the  second  and  fourth  the  conse- 
quents ;  and  the  last  is  said  to  be  a  fourth  proportional  to  the 
other  three  taken  in  order.  Thus,  in  the  last  proportion,  A 
and  C  are  the  antecedents,  and  B  and  D  the  consequents. 

4.  Three  quantities  are  in  proportion  when  the  first  has  the 
same  ratio  to  the  second,  that  the  second  has  to  the  third  ; 
and  then  the  middle  term  is  said  to  be  a  mean  proportional 
between  the  two'other.     For  example, 

3   :  6   ::  6   :   12  ; 
and  6  is  a  mean  proportional  between  3  and  12. 

5.  Quantities  are  said  to  be  in  proportion  by  inversion,  of 
inversely,  when  the  consequents  are  made  the  antecedents  and 
the  antecedents  the  consequents. 

Thus,  if  we  have  the  proportion 

3    :     6     ::     8     :     16. 

thfi  inverse  proportion  would  be 

6     :    3     ::     16     :    8. 


BOOK      III 


Of    Ratios    and    Proportions. 


6.  Quantities  are  said  to  be  in  proportion  by  allernattcfi,  oi 
altcrnateli/,  when  antecedent  is  compared  with  antecedent  and 
consequent  with  consequent 

Thus,  if  we  have  the  proportion 

3     :     6     :  :     8     :     IG, 
tlic  ahernate  proportion  would  be 

3     :     8     :  :     G     :     10. 

7.  Quantities  are  said  to  be  in  proportion  by  CLfnposition, 
when  the  sum  of  tlie  antecedent  and  conse(iucnl  is  compared 
cither  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

2  :     4     :  :     8     :     IG, 
the  proportion  by  composition  would  be 

2-f4     ;     4     ;:     8-}-lG     :     IG; 
that  is,  6     :     4     :  :     24     :     IG. 

8.  Quantities  are  said  to  be  in  proportion  by  division^  when 
the  difference  of  the  antecedent  and  consequent  is  compared 
either  wi^tli  llie  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

3  :     9     :  :     12     :     36, 
tlie  proportion  by  division  will  be 

9  —  3     :     9     ::     3G  — 12     :     36; 
Ihal  is,  G     :     9     :  :     24     :     36. 

9.  Equimultiples  of  two  or  more  quantities  are  the  products? 
which  arise  from  nudliplying  the  quantities  by  the  same 
number. 

Thus,  if  we  have  any  two  numbers,  as  6  and  5  and  multiply 


72  GEOMETRY 


Of    Ratios    and     Proportions 


them  both  by  any  number,  as  9,  the   equimultiples  will  be  54 
find  45  ;  for 

6x9  =  54         and         5x9  =  45. 
.Alio,  mxA   and  mxB   are    equimultiples   of  A  and  5, the 
common  multiplier  being  m, 

10.  Two  variable  quantities,  A  and  B,  are  said  to  be  re- 
ciprocalbj  2)'>'oportional^  or  inversely  proportional,  when  one 
increases  in  the  same  ratio  as  tlie  other  diminishes.  When 
this  relation  exists,  either  of  them  is  equal  to  a  constant 
quantity   divided   by   the   other. 

Thus,  if  we  had  any  two  numbers,  as  2  and  4,  so  related 
to  each  other  that  if  we  divided  one  by  any  number  we  must 
multiply  the  other  by  the  same  number,  one  would  increase 
in  the  same  ratio  as  the  other  would  diminish,  and  their 
product  would  not  be  changed. 

THEOREM    I. 

//  four  quantities  are  in  proportion,  the  product  of  the  two  ea 

trcmcs  will  be  equal  to  the  product  of  the  two  rrteana 

If  we  have  the  proportion 

A     :     B     '.'.     C     ',     D 

wo  have,  by  Def.  2, 

Il_D 
A~  C 
Uid  by  clearing  the  equation  of  fractions,  we  have 
BC^AD 
Sch    The   general   principle  is  verified   in    the   proportion 
between  the  numbers 

2     :      10     :  :      12     :     00 

which  gives 

2>60=:10x\2  =  120 


B  O  O  K     I  I  1 .  78 


Of     Ratioa    and    Proportions. 


THEOREM      II. 
If  four  quantities  are  so  related  to  each  other ^  that  the  product 
of  ttro  of  them  is  equal  to  the  product   of  the  other  two ;  thm 
rxto  of  them   may   be    made   the  means,   and   the   other   txDO   the 
CM.trcines  of  a  proportion. 

Let  -4,  i?,  C,  and  D,  have  such  vahics  that 

BxC=AxD 
Divido  both  sides  of  the  equation  by  A   and  we  have 

A 
Then  divide  both  sides  of  the  last  equation  by  C,  and  we 
have 

B_D 

7v~c 

hence,  by  Dcf.  2,  we  have 

A     :      B     :i      C     '.     D. 

Sch.  The  general  truth  may  be  verified  by  the  numbers 
2x18  =  9x4 
which  give 

2     :     4     :  :     9  18 

THEOREM     III. 

y  three   quantities    are   in   proportion,  the  product  of  the    two 
extremes  will  he  equal  to  the  square  cf  the  middle  term. 

Let  us  suppose  that  we  have 

A     :     B     .  :     B     :      C 
Then,  by  Def.  2,  we  have 

B_C 
A~  B 

an«l  by  clearing  the  equation  of  its  fractions,  vrt  bave 

'7 


74  GEO  M  E  T  R  V  . 


Of    Ratios    and    Prop  or  t  ione. 


Sch.  The  proposition  may  be  verified  by  the  numbers 
3     :     6     :  :     6     :     12 
which  give 

3x12  =  6x6=36 

THEOREM    IV. 

Ij  four  quantities  are  in  proportion,  they  will  he  in  proportion 
when  taken  alternately. 

Let  A     :     B     '.  :     C     '.     D 

Then,  by  Def.  2,  we  have 

B_B 

A~C 

Multiplying  both  members  of  this  equation  by  — ,  we  have 

B 

C_D 
A~B 
and  consequently, 

A     I     C     ::     B     :     D. 

Sch.  The  theorem  may  be  verified  by  the  proportion 
10     :      15     :  :     20     :     30 
for,  we  have,  by  alternation, 

10     :     20     :  :     15     :     30. 

THEOREM    V. 

[f  thcie  be  two  sets  of  proportions,  having  an  antecedent  and 
a  wnsequent  in  the  one,  equal  to  an  antecedent  and  a  consequent 
in  the  other;  then,  the  remaining  terms  will  he  proportional. 

If  we  have 
^     :     ^     :  .     C     .     A    and      A     :      B        :     E     :     F . 
then  we  shall  have 


B  O  O  K     I  I  I  76 


O  f    Ratios    and    Propor    iona. 

B     D  B    F 

A=C         ^"^  A  =  E 

Hence,  by  Ax.  1,  we  have 

D_F 

C~E 

and  consequcnily, 

C     :     D     ::     E     '.     F 

Sch.  The   proposition   may  bo   verified   by  the   following 

proportions, 

2     :     6     :  :     8     :     24        and       2     :     6     :  :      10     :     30 

which  give 

8     '     24     :  .     10     :     30. 

THEOREM    VI. 

J/ four  quantities  are  in  proportion,  tlie.y  will  be  in  proportion 
when  taken  inversely. 

If  we  have  the  proportion 

A     :     B     :.     C     '.     D 

we  have,  by  Th.  I, 

AxD=zBxC, 

or  BxCz=AxD. 

Hence,  we  have,  by  Th.  II, 

B     '.     A     '.:     D     :     C. 

Sch.  The  proposition  may  be  verified  by  the  proportion 
7     :      14     :  :     8     :     16; 
which,  when  taken  inversely,  gives 

14     :     7     :  :      16     :     8. 

THEOREM    VII. 

sffouT  t^antities  are  in  proportion^  they  wUl  be  in  proportion  oy 
composition. 


76  GEOMETRY 


Of    Ratios    and    Propoi    ion  a. 


Let  US  suppose  that  we  have 

A     .     B     :  :     C     :     D 
we  shaL  then  have 

AxD  =  BxC. 
To  each  of  these  equals,  add  BxD,  and  we  have 
(A+B)xD={C-\-D)xB; 
and  by  separating  the  factors  by  Th.  II,  we  have 
A-{-B     :     B     ::     C+D     :     D. 

Sch.  The    proposition    may  be   verified  by  the   following 
proportion, 

9     :     27     :  :     16     :     48. 
We  shall  have,  by  composition, 

9+27     :     27     :  :     16+48     :     48, 
that  is,  36     :     27     :  :     64     ;      48 

in  which  the  ratio  is  three  fourths. 

THEOREM    YIII. 

If  four  quantities  are  m  proportion^  they  will  he  in  proportion  by 
division. 

Let  us  suppose  that  we  have 

A     :     B     ','.     C     :     D, 
we  shall  then  have 

AxD  =  BxC. 
From  each  of  these   equals  let  us  subtract  BxD,  and  we 
have 

{A-B)xDr=z{C-D)xB; 
and  by  separating  the  factors  by  Th.  II,  we  have, 
A-B     :     B     '.  :     C-D     :     D. 

Sck    The  proposition  may  be  verified  by  the  proportion, 
24     •     8     : :     48     :      16 


{ 


BOO  K      I  I  I.  77 


Of    Ratios    and    Proportions 


We  have,  by  division, 

24-8     :     8     :  :     48-16     :     16; 
that  i8,  16     :     8     :  :     32     :     10; 

in  which  the  ratio  is  one-half. 

THEOREM     IX. 

Eqiial  multiples  of  two  quantities  have  the  same  ratio  aa  Uu. 
quantities  themselves. 

It  wo  have  the  proportion 

A     :     B     ".     C    I     D 
we  shall  ha\e 

B_D 
a"  C 

Now,  let  M  be  any  number,  and  by  it  multiply  the  iiu« 
merator  and  denominator  of  llie  first  member  of  the  equation 
which  will  not  change  its  value :  we  shall  then  ha\e 

M^BD 
UxA"  C 
and  her.ce  we  have 

'    My.A     '.     MxB     ::     C     :     D, 
that  is,  the  equal  multiples  Mx  A  and  MxB,  have  the  same 
ratio  as  A  to  B. 

Sch    The  proposition  may  be  verified  by  the  proportion, 
5     :      10     ::     12     •     24; 

for,  by  multiplj-ing  the  first  antecedent  and  consequent  by  any 
number,  as  6,  we  have 

30     :     60     :  :     12     :     24, 
V   ^vhich  the  ratio  is  still  2. 


78  GEOMETRY. 


Of    Ratios    and    Proportions. 


THEOREM    X. 

Ij  four  quantities  are  proportional^  and  one  antecedent  and  iLs 
wnsequnht  be  augmented  by  quantities  which  have  the  same  ratio 
as  the  antecedent  and  consequent,  the  four  quantities  will  stUl  he 
in  proportion 

liCt  us  take  the  proportions 

A     X     B     X  :     C    I     D,     2iii^     A     I     B     I  :     E     \     F, 

which  give 

AxD-BxC  and  AxF=zBaE; 

adding  these  equals  we  have 

Ax{D-\-F)  =  Bx{C-{-E); 
and  by  Th.  II,  we  have 

A     :     B     '.:     C^E     :     D+F 

in  which  the  antecedent  C  and  its  consequent  D,  are  augmeni- 

ed  by  the  quantities  E  and  F,  which  have  the  same  ratio. 

Sch.  The  proposition  may  be  verified  by  the  proportion, 

9     :     18     ;  :     20     :     40, 

in  which  the  ratio  is  2. 

If  we  augment  the  antecedent  and  its  consequent  bv  15  and 
30,  which  have  the  same  ratio,  we  have 

9     :     18     :  :     20+15     :     404  30 

that  is,  9     :     18     :  :     35     •     70, 

in  ^^hjch  the  ratio  is  still  2. 

THEOREM    XI. 

If  four  quantities  are  proportional,  and  one  antecedent  and  its 
consequent  be  diminished  by  quantities  which  have  the  same  ratio 
as  the  antecedent  and  consequent,  the  four  quantities  unll  still  bg 
in  pvportion 


B  0  O  K     I  I  I  .  79 


Of    Ratios    and    Proportiona. 


Let  us  lake  the  proportions 
A     :     B     :  :     C     :     D,     Bind      A     :     B     :  :     E     :     F. 
which  give 

AxD=BxC        and         AxF=BxE. 
By  subtracting  these  equalities,  we  have 
Ax{D-F)  =  Bx(C-E)i 
and  by  Th.  II,  wc  obtain 

A     :     B     :  :     C-E     :     D-F, 
in  which  the  antecedent  and  consequent,  C  and  Z>,  are  dimin- 
ished by  E  and  P,  which  have  the  same  ratio 

Sch.  The  proposition  may  be  verified  by  tlie  proportion, 
9     :     18     :  :     20     :     40, 
for,  by  diminisliing  the  antecedent  and  consequent  by  15  and 
30,  we  have 

9     :     18     :  :     20-15     :     40-30; 

that  is  9     :     18     :  :     5     :     10 

in  which  the  ratio  is  still  2. 

THEOREM    XII. 

Jf  wc  have  several  sets  of  proportions^  having  the  same  ratio,, 
any  antecedent  unll  he  to  its  consequent,  as  the  sum  of  the  onto 
cedents  to  the  sum  of  the  consequents. 

If  we  have  the  several  proportions, 

D  which  gives  AxD=BxC 
F  which  gives  A  x  F=Bx  E 
H     which  gives   A  x  H=Bx  O 

We  shall  then  have,  by  addition, 

Ax[D-]-F+H)=:Bx(C-rE+G)', 
and  consequently,  by  Th  II. 

A     :     B     ;:     C-\-E-\-G     :     D-^F-^Il. 


A 

B     : 

,     C 

A 

■     B     : 

:     E 

A 

B    ' 

:     G 

80  GEOMETRY. 


Of    Ratios    and    Proportions 


Sch.  The    proposition   may  be   verified   by  the   following 
proportions  :  viz. 
2     •     4     :  :     G     ;     12         and         1     ;     2     :  :     3     :     6 

Then,  2:4::     6-f-3     :     12-f6; 

that  is,  2     :     4     :  •     9     :     18, 

in  which  the  ratio  is  still  2. 

THEOREM    XIII. 

If  four  quantities  are  in  proportion,  their  squares  or  cubes  will 
also  be  proportional. 

If  we  have  the  proportion 

A     :     B     ::     C     :     D, 
it  gives 

n_D 

A~C 
Then,  if  we  square  both  members,  we  have 

and  if  we  cube  both  members,  we  have 
B^     D^ 

and  then,  changing  these  equalities  into  a  proportion,  we  have 
for  the  first, 

A^     :     B'     ::     C^     :     23^ 
and  foi  the  second 

A^  B^     :  •     C^     :     D 

Soh.  We  may  verify  the  proposition  by  the  proportion, 
2     :     4     :  :     6     :     12, 
and  by  squaring  each  term  we  have, 

4     :      IG     :  .     36     •      144 


BO  O  K     1  I  I  .  81 


Of    Ratios    and     Proportions. 


numbers  which  are  still  proportional,  and  in  which  the  ratio 
is  4. 

If  we  cube  the  numbers  we  have, 


2'     :     4' 

:•     6= 

•     12^ 

tiiat  is,               8     :     64     :  ' 

2.6     • 

1721 

in  wliich  the  ratio  is  8. 

THEOREM    XIV. 

If  we  have  two  sets  of  proportional  quantities,  the  products  oj 
the  corresponding  terms  will  be  proportional. 

Lot  U8  take  the  proportions, 
A     '     B     :  :     C     :     D        which  gives 


B_D 
A~C 
F_H 
E~G 


E     :     F    :  :     G     :     H       which  gives 

Multiplying  the  equalities  together,  we  have 

BxF     DxH 
AxE^CxG 
md  this  by  Th.  II,  gives 

AxE     :     BxF    ::     CxG     :     DxH. 
Sch.    The   proposition   may  be  verified   by  the  followmg 
proportions : 

'8     ;     12     :  :     10     :     15, 
and  3     :       4     :  :       6     :       8 ; 

we  shftll  then  have 

24     :     48     :  :     60     :  120 
whi'jh  are  proportional,  the  ratio  being  tK 


GEOMET  R  Y. 


BOOK    IV 

01^  THE  MEASUREMENT  OF  AREAS,  AND  THB 
PROPORTIONS  OF  FIGURES. 

DEFINITIONS. 

1  Similar  figures,  are  those  which  have  the  angles  of  tlie 
one  equal  to  the  angles  of  the  other,  each  to  each,  and  the 
sides  about  the  equal  angles  proportional. 

2.  Any  two  sides,  or  any  two  angles,  which  are  like  placed 
in  the  two  similar  figures,  are  called  homologous  sides  oi 
angles. 

3.  A  polygon  which  has  all  its  angles  equal,  each  to  each, 
and  all  its  sides  equg,l,  each  to  each,  is  called  a  regular  polygon. 
A  regular  polygon  is  both  equiangular  and  equilateral. 

4.  If  the  length  of  a  line  be  computed  in  feet,  one  foot  is 
the  unit  of  the  line,  and  is  called  the  linear  unit.  Il  the  length 
of  a  line  be  computed  in  yards,  one  yard  is  the  linear  unit 

5.  If  we  describe  a  square  on  the  unit 
of  length,  such  square  is  called  the  unit  of 
surface.  Thus,  if  the  linear  unit  is  one 
foot,  one  square  foot  will  be  the  unit  of 
surface,  or  superficial  unit. 


BOOK     IV. 


83 


Of    Parallelograms, 


lyd.-3  1t«t. 

6.  If  the  linear  unit  is  one  yard,  one 
square  yard  will  be  the  unit  of  surface  ; 
and  this  square  yard  contains  nine  square 
feet. 


7.  The  area  of  a  figure  is  the  measure  of  its  surface.  The 
unit  of  tlie  number  which  expresses  the  area,  is  a  square,  the 
side  of  wliich  is  the  unit  of  length. 

8.  Figures  have  equal  areas,  when  they  contain  the  same 
measuring  unit  an  equal  number  of  times. 

9.  Figures  which  have  equal  areas  are  called  equivalent. 
The  term  equal,  when  applied  to  figures,  implies  an  equality 
in  all  respects.  The  term  equivalent,  implies  an  equality  in 
one  respect  only  :  viz.  an  equality  in  their  areas.  The  sign 
«0=,  denotes  equivalency,   and  is  read,  is  equivalent   to. 


THEOREM    I. 

Parallelograms  which  have  equal  bases  and  equal  altitudes,   arc 

equivalent. 

Place  the  base  of  one  parallel- 
ogram on  that  of  the  other,  so  that 
AB  shall  be  the  tommon  base  of 
the  two  parallelograms  ABCD 
and  ABEF.  Now,  since  the  par- 
allelograms have  tlic  same  altitude,  their  upper  bases,  DC  and 
FE,  will  fall  on  the  same  line  FEDC,  parallel  to  AB.  Since 
the  opposite  sides  of  a  parallelogram  are  equal  to  each  other 
(Bk.  I  Th.  jaiii),  AD  is  equal  to  BC.  Also,  DC  and  FE  are 
each  equal  to  AB :  and  consequently,  they  are  equal  to  each 


;^4  G  E  O  .'\l  E  T  R  Y . 

Of   Triangles    and   Parallelogramj. 

olhei  (Ax.  1 ).     To  each,  add  ED :     p ^       p  ^. 

then  will  CE  bo  equal  to  DF.  \~  ^^7"  V 

But  since  the  line  FC  cuts  the  \  /\  / 

tw'O    parallels    CB   and  DA,   the  \Z___a/ 

angle  BCE  will  be   equal  to  the  -^  ^ 

angle  ADF  (Bk.  I.  Th.  xiv) :  hence,  the  two  triangles  ADF 
and  BCE  have  two  sides  and  the  included  angle  of  the  one 
equal  to  two  sides  and  the  included  angle  of  the  other,  each 
to  each ;  consequently,  they  are  equal  (Bk.  I.  Th.  iv). 

If  then,  from  the  whole  space  ABCF  we  take  away  the  tri- 
angle ADF,  there  will  remain  the  parallellogram  ABCD ;  but 
if  we  take  away  the  equal  triangle  BEC,  there  will  remain  the 
parallelogram  ABEF:  hence,  the  parallelogram  ABEF  is 
equivalent  to  the  parallelogram  ABCD  (Ax.  3). 

Cor.    A   parallelogram    and    a            |    7  j    7 

rectangle,  having  equal  bases  and              / 
equal  ahitudes,  are  equivalent  V. 


THEOREM    II, 

Triangles    which    have   equal   bases   and    ".qual  altitudes,    an 
equivalent. 

Place  the  base  of  one  triangle    F D,_E C 

on  that  of  the  other,  so  that  ABC     \         \\/    ^^ 

and  ABB    shall    be    iv?o   trian-  \     /  /0\      / 

jjrles,  having  a  common  base  AB,  y^^— -\\ 

°  °  .  A  B 

and  for  their  altitude,  the  distance 

between  the  two  parallels  AB,  FC :  then  will  the  triangle 

ABC  be  equivalent  to  the  triangle  ADB. 

For,  through  A  draw  AE  parallel  to  BC,  and  AF  parallel  to 

SD,  formiii^  the   two  parallelograms  BE  and   BF     Then 


BOOK    IV 


85 


(Jf    Triangles    and    Parallelograms. 

since  these  parallelograms  have   a  common  base  and  equal 
altitudes,  they  will  be  equivalent  (Th.  i). 

But  the  triangle  ABC  is  lialf  the  parallelogram  BE  (Bk.  1 
Th.  xxiii);  and  >1j5D  is  half  the  equal  parallelogram  BF. 
hence,  the  triangle  ABC  Is  equivalent  to  the  triangle  ABD, 


THEOREM    111. 

If  a  triangle  and  a  parallelogram  have  equal  bases  and  equal 
altitudes^  the  triangle  xcill  be  half  the  parallelogram. 

Place  the  base  of  the  triangle  on  the 
base  of  the  parallelogram,  so  tliat  AB 
shall  be  the  common  base  of  the  tri- 
angle and  parallelogram :  then  will  the 
triangle  ABE  be  half  the  parallelogram 
BD. 

For,  draw  the  diagonal  AC  Then,  since  the  altitude  of 
the  triangle  AEB  is  equal  to  tha^  of  the  parallelogram,  th« 
vertex  will  be  found  some  where  in  CD^  or  in  CI)  produced. 
Now  the  two  triangles  ABC  and  ABEj  having  the  same  base 
AB,  and  equal  altitudes,  are  equivalent  (Th.  ii).  But  the  tri- 
angle ABC  is  half  the  parallelogram  BD  (Bk.  I.  Th.  xxiii)  : 
hence,  the  triangle  ABE  is  half  the  parallelogram  BD  (Ax.  i\. 

Cor.  Hence,  if  a  trijlngle  and  a  rect- 
angle have  equal  bases  and  equal  alli- 
uides,  the  triangle  will  be  half  the 
rectangle. 

For  the  rectangle  would  be  equiva- 
lent to  a  parallelogram  of  the  same  base 

and  altitude  (Th.  i.  Cor.),  and  since  the  triangle  is  half  the 
parallelogram,  it  is  also  equivalent  to  half  the  rectnnsile 


8G 


GEOMETRY 


Of    Rectansloa. 


D        C     . 

"—^ 

/' 

/ 

/ 

/ 

/ 

' 

/ 

/ 

\-4  ^-i 

THEOREM    IV. 

Rcctar,gles  which  are  described  on  equal  lines  are  equivalent 

liCt  BD  and  FHhe  two  rectangles, 
having  the  sides  AB,  BC,  equal  to 
the  two  sides  £F,  FG,  each  to 
each:  then  will  the  rectangle  ABCD^ 
described  on  the  lines  AB,  BC,  be 
equivalent  to  the  rectangle  EFGH, 
described  on  the  lines  EF^  FG. 

For,  draw  the  diagonals  AC,  EG,  dividing  each  parallel 
ogram  into  two  equal  parts. 

Then  the  two  triangles,  ABC,  EFG,  having  two  sides  and 
the  included  angle  of  the  one  equal  to  two  sides  and  the  in- 
cluded angle  of  the  other,  each  to  each,  are  equal  (Bk.  1. 
Th.  iv).  But  these  equal  triangles  are  halves  of  the  respective 
rectangles  (Th.  iii.  Cor.) :  hence,  the  rectangles  are  equal 
(Ax.  7) ;  and  consequently  equivalent. 

Cor,  The  squares  on  tqual  lines  are  equal.  For  a  square 
is  but  a  rectangle  having  its  sides  equal. 


THEOREM     V. 


l^wc  rectangles  having  equal  altitudes  are  tc  each  other  as  theif 

bases. 


Let  AEFD  and  EBCF  be  two 
rectangles  having  the  common  alti- 
tude AD ;  then  will  they  be  to  each 
other  as  the  bases  AE  and  EB. 


D 

F             V 

1 

\ 

1      1 

i    1 

A 

1 

5      '5 

For,  suppose  the  base  ^^  to  be  to  the  base  EB,rs  any  two 
numbers,  say  tl  e  numbers  4  and  3      T>et  AE  be  then  divided 


B  0  O  K     1  V  .  87 


Of    Rectangles. 


into  four  equal  parts,  and  EB  into  three  equal  parts,  and 
tlirougli  the  points  of  division  draw  parallels  to  AD  We 
Bl'.all  thus  form  seven  rectangles,  all  equivalent  to  each  other 
since  they  have  equal  bases  and  equal  altitudes  (Th.  iv). 

But  the  rectangle  AEFD  will  contain  four  of  these  partial 
rectangles,  while  the  rectangle  EBCF  will  contain  three  ; 
hence,  the  rectangle  AEFD  will  be  to  the  rectangle  EBCF  as 
4  to  3  ;  that  is,  as  the  base  AE  to  the  base  EB. 

The  same  reasoning  may  be  applied  to  any  other  rect^ 
angles  whoso  bases  are  whole  numbers :  hence, 

AEFD    :     EBCF    :  i     AE     i     EB. 

THEOREM    VI. 

Any  two  rectangles  are  to  each  other  as  the  products  of  thtit 
bases  and  altitudes. 


l.et  ABCD  and  AEGF  be        H D 


two  rectangles  :  then  will 
ABCD  :  AEGF  •  :  ABxAD 
:  AFxAE 

For,  having   placed  the    two 
rectangles    so    that   BAE    and        G 
DA  F  shall  form  straight  lines,  produce  the  sides  CD  and  GE 
until  they  meet  in  H. 

Then,  the  two  rectangles  ABCD^  AEHD,  having  the  com- 
mon altitude  AD,  are  to  each  other  as  their  bases  AB  and 
AE  (Th.  v).  In  like  manner,  the  two  rectangles  AEHD 
AEGF,  having  the  same  altitude  AE,  arc  to  each  other  w 
their  bases  AD  and  AF.     Thus,  we  have  the  proportions 

ABCD     :     AEHD     :  :     AB     :     AE, 
AEHD     :     AEGF     :  ;      AD     :     AF, 


6S> 


GEOMETKY. 


Of  Bectangles 


If,  now,  we  multiply  the  cor- 
responding terms  together,  the 
products  will  be  proportional 
(Bk.  III.  Th.  xiv.)  ;  and  the 
common  multiplier  AEHD  may 
be  omitted  (Bk.  III.  Th.  ix.)  : 
hence,  we  shall  have 


H 


D 


G 


ABCD 


AEGF 


ABXAD    :     AExAF, 


Sch,  Hence,  the  product  of  the  base 
by  the  altitude  may  be  assumed  as  the 
measure  of  a  rectangle.  This  product 
will  give  the  number  of  superficial  units 
in  the  surface :  because,  for  one  unit  in 
weight,  there  are  as  many  superficial  units 
as  there  are  linear  units  in  the  base ;  for  two  units  in  height, 
twice  as  many;  for  three  units  in  height,  three  times  as 
many,  &c. 


THEOREM    VII. 


The  sum  of  the  rectangles  contained  by  one  line^  and  tht 
several  parts  of  another  line  any  way  divided^  is  equivalent  to  tJu 
rectangle  contained  by  the  two  whole  lines. 

Let  AD  be  o  e  line,  and  AB  the  other, 
divided  into  the  parts  AE^  EF^  FB :  then 
will  the  rectangles  contained  by  AD  and 
AE,  AD  and  EF,  AD  and  FB,  be  equiv- 
alent to  the  rectangle  A  C  which  is  con- 
tained by  the  lines  AD  and  AB, 

For,  through  the  points  E  and  F  draw  the  lines  EG  and 
FII,  parallel  to  the  line  AD  :    then  will  the  rectangle  AO 


D      G       H      C 

1      i 

;    i 

'    b 

B  0  0  K     I  V  .  89 


Of    Areas    of     Parallelograms, 


be  equal  to  the  rectangle  of  ADxAE  ;  EH  will  be  equal  to 
EGxEF,oTtoADxEF;  and  FC  willhe  equal  io  FHxFB 
or  to  AD  X  FB. 

But  the  rectangle  ^iCis  equal  to  the   sum  of  tbe   partinJ 
rectangles :  hence, 

ADxAD=0=ADxAE-{-ADxEF+AD>.Fn. 

THEOREM    Mil. 

The  area  of  any  parallelogram  is  equal  to  the  product  of  its  base 

by  its  altitude. 

Let  ABCD  be  any  parallelogram,  and 

BE  its   altitude  :    then  will  its  area  be       f^ T~7' 

equal  io  ABx BE. 

For,    draw    AF  perpendicular   to   the 
base  AB,  and  produce  CD  to  F.     Tlien, 


the  parallelogram  BD  and  the  rectangle  BF^  having  the  samr 
base  and  altitJide  are  equivalent  (Th.  i.  Cor.).  But  the  arei 
of  the  rectangle  BF  is  equal  to  the  product  of  its  base  AB  h^ 
the  altitude  AF  (Th.  vi.  Sch.):  hence,  the  area  of  the  paral 
lelogram  is  equal  to  ABx  BE. 

Cor.  Parallelograms  of  equal  bases  are  to  each  other  as  then 
altitudes ;  and  if  their  altitudes  are  equal,  they  arc  to  each 
other  as  their  bases. 

For,  let  B  be  the  common  base,  and  C  and  D  the  altitudes 
of  two  parallelograms.  'I'hen,  by  the  theorem,  theii  areas  arc 
to  each  other,  as 

B\C     :     BxD, 
that  is   (Bk.  III.  Th  ix),  as  C  :  D 

If  A  and  B  be  their  bases,  and  C  their  common  altitude, 
then  they  wM  be  to  each  other  as 

AxC     :     BxC:  that  is,  as  A     :     F 


90 


G  E  O  1\I  E  T  R  r  . 


Areas    cf    Triangles    and    Traperoids. 


THEORExM    IX 

The  area  of  a  triangle  is  equal  to  half  ii^  product  of  its  base  hi. 
its  altitude. 

Let  ABC  ha  any  triangle  and  CD  its 
altitude :  then  will  its  area  be  equal  to 
half  the  product  of  AB  x  CD. 

For,  through  B  draw  BE  parallel  to 
A  C,  and  through  C  draw   C£  parallel 
to  ^Z? ;  we  shall  then  form  the  parallelogram  AE,  having  the 
same  base  and  altitude  as  the  triangle  ABC. 

But  the  area  of  the  parallelogram  is  equal  to  the  product  of 
the  base  ABhy  its  altitude  DC ;  and  since  the  parallelogram  is 
■double  the  triangle  (Th.  iii),  it  follows  that  the  area  of  the  tri 
angle  is  equal  to  half  this  product :  that  is,  to  half  the  product 
o(  ABxCD 

Cor.  Two  triangles  of  the  same  altitude  are  to  each  othei 
as  their  bases  ;  and  two  triangles  of  the  same  base  are  to  each 
other  as  their  altitudes.  And  generally,  triangles  are  to  each 
other  as  the  products  of  their  bases  and  altitudes. 


THEOREM      X. 


7  he  area  of  a  trapezoid  is  equal  to  half  the  product  of  its  altitud* 
multiplied  by  the  sum  of  its  parallel  sides. 

Let  ABCD  be  a  trapezoid,  CG 
its  altitude,   and  AB,  DC  its  par- 
allel   sides :    then  will  its   area  be 
equal  to  half  the  product  of 
CGx{AB+DC) 


ROOK     IV. 


91 


Of    Rectangles. 


For,  produce  AB  iiniil  BE  is  ecjutil  to  DC,  and  complete 
Jlio  rectangle  AF ;  also,  draw  BII  perpendicular  to  AD. 

Then,  the  rectangle  AC  will  be  equivalent  to  BF,  since  they 
have  equal  bases  and  equal  altitudes  (Th  iv).  The  diagonal 
BC  viill  divide  the  rectangle  Gil  into  two  equal  triangles; 
and  hence,  the  trapezoid  A  BCD  will  be  equivalent  to  the 
trapezoid  BEFC ;  and  consequent ly,  the  rectangle  AF,  is 
double  the  trapezoid  ABCD. 

But   the   rectangle   AF  \s   equivalent    to   the    product  of 
ADxAE;  that  is,  to   CGx{AB-{-DC)\  and  consequently 
the  trapezoid  ABCD  is  equal  to  half  that  product. 


THEOREM    XI. 


If  a  line  he  divided  into  two  parts,  the  square  described  on  the 
whole  line  is  equivalent  to  the  sum  of  the  squares  described  on  the 
two  parts,  together  with  twice  the  rectangle  cnntaijicd  by  the  parts 


Let  the  line  AB  he  divi'led  into  iv/o 
parts  at  the  point  E:  then  wnl  the  scjuarc 
described  on  ABhQ  equivalent  to  the  two 
squares  described  on  AE  and  EB,  to- 
gether with  twice  the  rectangle  contained 
by  AE  and  EB :  that  is 


JjL 


^^ 


E 


R 


AEi' 


■AE^-\-EB'^^2AExEB. 


Foi  let  AC  be  a  square  on  A  B,  and  AF  a  square  on  AE 
and  produce  the  sides  EF  and  GF  to  //  and  /. 

Then  since  EH  is  equal  to  AD,  being  the  opposite  side  o( 
a  rectangle,  it  is  also  equal  to  AB ;  and  GI  is  likewise  equai 
u^  AB      If  therefore,  from  these  equals  we  take  av^  ay  EF  and 


92 


GEOMETRY 


Of    Rectangles 


GF,  there  will  remain  FH  equal  to  FI, 
and  each  will  be  equal  to  HC  or  IC ;  and 
since  the  angle  at  i^  is  a  right  angle,  it 
follows  that  FC  is  equal  to  a  square  de- 
scribed on  EB.  It  also  follows,  that  DF 
and  FB  are  each  equal  to  the  rectangle 
of  AE  into  EB, 


D 


H 


B 


But  the  square  ABCD  is  made  up  of  four  parts,  viz.,  the 
square  on  AE ;  the  square  on  EB ;  the  rectangle  DF ,  and 
the  rectangle  FB.  Hence,  the  square  on  AB  is  equivalent 
to  the  square  on  AE  plus  the  square  on  EB,  plus  twice  the 
rectangle  contained  by  AE  and  EB. 

Cor.  If  the  line  AB  be  divided  into 
two  equal  parts,  the  rectangles  DF  and 
FB  would  become  squares,  and  the  square 
described  on  the  whole  line  would  be 
equivalent  to  four  times  the  square  de- 
scribed on  half  the  line. 


Sch.  The  property  may  be   expressed  in  the  language  of 
algebra,  thus, 


THEOREM      XII. 

7 Vie  square  acscrihcd  on  the  hypothenuse  of  a  right  angleJ 
triangle,  is  equivalent  to  the  sum  of  the  squares  described  on  tht 
other  two  sides. 


BOOK    17. 


93 


Of    Right    Angled    Triangles. 

Let  BA  C  be  a  right  an- 
gled triangle,  right  angled  at 
A:  tlicn  will  the  square  dc- 
Bcribcd  on  the  hypothenuse 
BCj  be  equivalent  to  the  two 
squares  described  on  I^A 
and  AC. 

Having  described  the 
squares  £Gy  JJL,  and  AIj 
let  fall  from  A,  on  the  hy- 
pothenuse, the  perpendicular 
AD,  and  produce  it  to  JS;  then  draw  the  diagonals  AF,  CIT. 

Now,  the  angle  ABF  is  made  up  of  the  right  angle  FBC 
and  the  angle  CBA ;  and  the  angle  CBII  is  made  up  of  the 
right  angle  ^i?//and  the  same  angle  CBA:  hence,  the  angle 
ABF  is  equal  to  CBII.  But  FB  is  equal  to  BC,  being  sides 
of  the  same  square;  and  for  a  like  reason,  BA  is  equal  to 
HB.  Therefore,  the  two  triangles  ABF  and  CBH,  having 
two  sides  and  the  included  angle  of  the  one  equal  to  two  sides 
and  the  included  angle  of  the  other,  each  to  each,  are  equal 
(Bk.  I.     Th.  iv). 

Since  the  angles  BAC  and  BAL  are  right  angles,  as 
Mso  the  angle  ABII,  it  follows  that  CAL  is  a  straight  line 
parallel  to  BH.  (Bk.  T.  Th.  ii.  Cor.  3).  Hence,  the  square 
HA  and  the  triangle  HBC  stand  on  the  same  base  and  be- 
tween the  same  parallels;  therefore  the  triangle  is  half  the 
tquare  (Th.  iii.  Cor.).  For  a  like  reason,  the  triangle  ABF 
is  half  the  rectangle  BE. 

But  it  has  already  been  proved  that  the  triangle  ABF  ia 
equal  to  the  triatigle  CBH :  hence,  the  rectangle  BE,  which 
is  double  the  former,  is  equivalent  to  the  square  BL,  which  ifl 
double  the  latter  (Ax.  6). 


94 


G  E  0  i\I  E  T  R  Y 


Of    Right    Angled    Tr' angles. 


In  the  same  manner  it 
may  be  proved,  that  the  rect- 
angle DG  is  equivalent  to 
the  square  CK 

But  the  two  rectangles 
DEy  DG,  make  up  the 
square  BG  :  therefore,  the 
square  BG,  described  on 
the  hypothenuse,  is  equiva- 
lent to  the  squares  BL  and 
CK,  described  on  the  other 
I  wo  sides. 

Cor.  Hence,  the  square  of  either  side 
of  a  right  angled  triangle  is  equivalent  to 
the  square  of  the  hypothenuse  diminijshed 
by  the  square  of  the  other  side.  That  is, 
in  the  light  angled  triangle  ABC 

ab'^oIc^-bc' 

or  BC^OAC^-AB^ 

Sck.  The  last  theorem 
may  be  illustrated  by  de- 
scribing a  square  on  the  hy- 
pothenuse BC,  equal  to  5, 
also  on  the  sides  BA,  AC, 
respectively  equal  to  4  and  3  ; 
and  observing  that  the  num- 
ber of  small  squares  in  the 
large  square  is  equal  to  the 
number  in  the  two  small 
squares 


C 


B  O  0  K     I  V  .  95 


Of    Triangle    Sides    cut    Proportionally. 

THEOREM    XIII. 

I)  a  line  be  drawn  parallel  to  the  base  of  a  tnangle,  it  will  divide 
the  other  two  sides  proportionally. 

Let  ABC  be  any  triangle,  and  DE  a 
firaight  line  drawn  parallel  to  the  base 
BC:  then  will 

AD     :     DB     :-     AE     :     EC. 

For,  draw  BE  and  DC.     Then,   ilie 
two  triangles  BDE  and  DCE  have  the 
same  base  DE,  and  the   same   altitude,     B 
since  tlieir  vertices   B  and  C,  lie  in  the  lini   BC  parallel  to 
DE :  hence,  they  are  equivalent  (Th.  ii). 

Again,  the  triangles  ADU  and  BDE,  ha\e  a  common  ver- 
tex  E,  and  the  same  altitude ;  consequently,  they  are  to  each 
other  as  their  bases   (Th.  ix.  Cor.)  ;  hence,  we  h»ve 
ADE     :     BDE     :  :     AD     :     DB. 

But  the  triangles  ADE  and  CDE,  having  a  common  vertex 

D,  are  to  each  other  as  their  bases  AE  and  EC :  hence,  we 

have 

ADE     :     CDE     :  :     AE     :     EC. 

But  the  triangles  BDE  and  CDE  have  been  proved  equiva- 
lent :  hence,  in  the  two  proportions,  the  first  antecedent  and 
consequent  in  each  are  equal :  therefore,  by  (Bk.  III.  Th  v) 

we  have 

AD     :     BD     :  :     AE     :     EC. 

Cor.  The  sides  AB,  AC,  are  also  proportional  to  ihe  part* 
AD,  AE,  or  to  BD,  CE. 

For,  by  composition  (Bk.  III.  Th.  vii),  we  have 

AD-\-BD     :     BD     ::     AE+EC     :     EC. 

TTien,  by  alternation  (Bk.  111.  Th.  iv). 
AB  :  AC  :  :  BD:  EC,  hence,  also,  AB  :  AC  :  :  AD  :  AE 


96 


GEOMETRY. 


Proportions   of  Triancrles, 


THEOREM    XIV. 

A  line  which  bisects  the  vertical  angle  of  a  triangle  divider 
the  base  into  two  segments  which  are  proportional  to  the  adjacent 
nde. 

Let  AC£  be  a  triangle,  hav- 
ing the  angle  C  bisected  by  the 
Hue  CD:  thet  will 
AD    :    r>0    ::    AG    :    CB. 

For,   draw   £E  parallel    to 
CD  and  produce  AC  to  E. 
Then,  since   CB  cuts  the  two 
parallels  CD,  EB,  the  alternate  angles  BCD  and   CBE  are 
equal  (Bk.  I.  Th.  xii)  :  hence,  CBE  is  equal  to  angle  ACD. 

Buf,  since  AE  cuts  the  two  parallels  CD,  BE,  the  angle 
ACD  is  equal  to  CEB  (Bk.  I.  Th.  xiv)  :  consequently,  the 
angle  CBE  is  equal  to  the  angle  CEB  (Ax.  I)  :  hence,  the 
bide  CB  is  equal  to  CE  (Bk.  I.  Th.  vii.) 

Now,  in  the  triangle  ABE  the  line  CD  is  drawn  parallel 
to  BE:  hence,  by  the  last  theorem,  we  have 
AD    :    DB    :  :    AC   :    CE, 
and  by  placing  for  CE,  its  equal  CB,  we  have 
AD    :    DB    ::    AC   :    CB. 

THEOREM    XV. 

Equiangular  triangles  have  their  sides  proportional,  and  are 


Let  ABC  and  DEFhe  two  equi- 
angular triangles,  having  the  angle 
A  equal  to  the  angle  D,  the  angle  C 
to  the  angle  F,  and  the  angle  B  to 
the  angle  E:  then  will 

AB    :    AC   ::    DE  :    DF 


B 


C  E 


BOOK      IV 


97 


Proportions    of    Triangles. 

For,  on  the  sides  of  the  larger  triangle  DEF,  make  Dl 
equal  to  -4C  and  DG  equal  to  AB,  and  join  IG.  'I'hen  tho 
two  triangles  ABC  and  DIG^  having  two  sides  and  the  in* 
oluded  angle  of  tlie  one  equal  to  two  sides  and  the  included 
angle  of  the  other,  each  to  each,  will  be  equal  (Hk.  I  Th.  iv) 
Hence,  the  angles  /  and  G  are  equal  to  C  and  />,  and  conso 
.]uently,  to  the  angles  F  and  E:  therefore,  /G  is  parallel  to 
£F(Bk.  I.  Th.  xiv,  Cor.  1). 

Now,  in  the  triangle  DEF,  since  IG  is  parallel  to  the  base, 
we  have  (TIi.  xiii). 

DG     :     DI     :  :     DE     :     DF, 
iliai  i.s,  AB     '     AC    ::     DE     :     DF. 


THEOREM     XVI. 

Two  triangles  which  have  their  sides  proportional  are  equiarir- 
gular  and  similar. 

Let  BAC  and  EDF  be  two 
triangles  l.a\  in^- 

BC  .  EF  :  :  AB  :  ED, 
and   BC  :  EF  :  :  AC  :  DF; 

then  will   they  have  the  cor  res-      ^ 
ponding  angles  eq'i:il,  \\7...  the  angle 

Bz^E,     A  =  D     and      C=F. 

For.  at  the  point  E  make  FEG  equal  to  tho  angle  B, 
find  at  F  make  the  angle  E  FG  equal  to  the  an^le  C .  Tlien 
will  the  angle  at  G  be  equal  to  A,  and  the  two  triangles  BAC 
and  EGF  will  be  equiangukr  (Bk.  I    Tli   xvii.  Cor  1). 

Tlicrcfore,  by  the  last  theorem,  we  shall  have 

BC     :     EF     ::    AB     :     EG  ; 
9 


98 


(5  E  O  M  E  T  R  \' 


Proportions    of    Triangles, 


but  by  hypothesis, 

BC  :  EF  ::  AB  :  DE: 

hence,  EG  is  equal  to  ED. 

By  the  last  theorem  we  also 
liave 

BC     :     EF     :: 
and  by  hypothesis, 

BC     :     EF    :  : 


AC 


AC 


DF, 


hence,  FG  is  equal  to  DF. 

Therefore,  ilie  triangles  DEF  and  EGF,  liaving  their  three 
sides  equal,  each  to  each,  are  equiangular  (Bk.  I.  Th.  viii). 
But,  by  construction,  the  triangle  EFG  is  equiangular  with 
BAC :  hence,  the  triangles  BAC  and  EDF  are  equiangular, 
and  consequently  they  are  similar. 

Sck.  By  Theorem  XV,  it  appears  that  if  the  corresponding 
angles  of  two  triangles  are  equal,  each  to  each,  the  correspond- 
mg  sides  will  be  proportional ;  and  in  the  last  theorem  it  was 
proved  that  if  the  sides  are  proportional,  the  corresponding 
angles  will  be  equal. 

Now,  these  proportions  do  not  hold  good  in  the  quadrilate- 
rals. For,  in  the  square  and  rectangle,  the  corresponding 
angles  are  c(]ual,  but  the  sides  are  not  proportional ;  and  the 
angles  of  a  parallelogram  or  quadrilateral,  may  be  varied  at 
pleasure,  witlioui  altering  the  lengths  of  the  sides. 


THEOREM    XVII. 


ij  two  triangles  have  an  angle  in  the  one  equal  to  an  angle  in 
the  other,  and  the  sides  containing  these  angles  propo?'tional,thf 
two  triangles  unll  be  eq'iiangttlar  and  similar. 


BOOK     IV. 


99 


Proportion  a    of    Ti  i angles 


Let  ABC  and  DEF  be  two  tri- 
angles having  the  angle  A  equal  to 
(ha  angle  D,  and 

Ali  DE     :  :     AC     ,     DF; 

ihon    will    ilic     two    triangles     be 
similar. 

For,  lay  olT^G  equal  to  DE,  and  through  G  draw  GfipaT' 
allel  to  BC.  Then  the  angle  AG  I  will  be  equal  to  the  angle 
ABC  (Bk.  I.  Th.  xiv) ;  and  the  triangles  AGI  smdABC  will 
be  equiangular.     Hence,  we  shall  have 

AB     :     AG     ::     AC     :     AL 

Out,  by  hypothesis,  we  have 

AB     '.     DE     '.'.     AC     \     DF, 

and  by  construction,  ^G  is  equal  to  DE;  therefore,  Al  u 
equal  to  DF,  and  consequently,  the  two  triangles  AGI  and 
DEF  are  equal  in  all  their  parts  (Bk.  1.  Th.  iv).  But  the  tri- 
angle ABC  is  similar  to  AGI,  consequently  it  is  similar  to 
DEF 


TIinOREM   -Win. 

ij  from  the  rigid  angle  of  a  right  angled  triangle,  a  perpfK- 
diculiir  be  let  fall  on  the  hi/polhcnuse,  then 

I.  The  two  partial  triangles  thus  formed  will  be  similar  ta 
fticA  other  and  to  the  whole  triangle. 

Ix.  Either  side  including  tJie  right  angle  will  be  a  mean  pro- 
pt>rtiorMl  between  the  hypothcnuse  and  the  adjacent  segment. 

III.  The  perpendicular  will  be  a  mean  proportional  hetuven  tkt 
segments  of  the  hypothmuse 


100 


<i  E  t)  M  E  T  P  Y 


Proportions    of    Triangleg. 


Let  ABC  be  a  right  angled 
triangle,  and  AD  perpendicular 
to  the  liypothemise. 

'Die  two  triangles  BAC  and 
BAD  having  the  common  angle 
6,  and  the  right  angle  BAC  equal 
to  the  right  angle  at  D,  will  be  equiangular  (13k  I.  Th.  xvii 
Cor.  1);  and,  consequently,  similar  (Th.  xv).  For  a  like 
reason  the  triangles  BAC  and  CAD  are  similar. 

Now.  from  the  triangles  BAC  and  BAD,  we  have 

BC     :     BA     :  :     BA     :     BD. 

From  the  triangles  BA  C  and  CAD,  we  have 
BC     :     CA     '.:     CA     :     CD: 
end  from  the  triangles  BAD  and  DA C,  we  have 
BD     :     AD     :  :     AD     '.     DC. 

Cor.  If  from  a  point  A,  in  the 
circumference  of  a  circle,  AD  he 
drawn  perpendicular  to  any  diam- 
eter as  BC,  and  the  chords  AB 
AC  hn  also  drawn,  then  the  an- 
gle BAC  will  be  a  right  angle 
(Bk.  II.  Th.  x):  and  by  the 
theorem  we  shall  have, 

1st  The  perpendicular  AD    a  mean  proportional  between 
the  setrments  BD  and  DC. 

2d  Kach  chord  will  be  a  mean  proportional  between  the 
diameter  and  llie  adjacent  segment. 

That  is,  Alf=BDxDC 

a1?=BCxBV 


AC^=BCxCD 


BOOK       i  V  . 


IGl 


Proportions    of    Trianglei 


TIIEOREBf     XIX. 


Similar  triangles  are  to  each  other  as  the  squares  described  on 
titeir  homologous  sides 

Lot  ABC  and  DBF  be 
tuo  similai  triangles,  and 
A  L  and  DN  the  squares  de- 
scribed on  the  liomologous 
Bides  AB,  DE:  then  will 
the  triangle 

ABC  :  DEF  :  :  AL  :  DN. 

For,  draw  CG  and  F// perpendicular  to  the  bases  AB,  DE. 
and  draw  the  diagonals  BK  and  EM. 

Then,  the  similar  triangles  ABC  and  DEF,  having  their 
homologous  sides  proportional,  we  have 

AC    '.     DF    '.'.     AB     '.     DE; 

and  the  two  ACG,  DFH,  give 

AC     :     DF    I'.     CG     :     FH:; 
hence,  (Bk.  111.  Th.  v),  we  have  . 

AB     :     DE     :  :     CG     i     fU. 
or  (Bk.  III.  Th.  iv), 

AB     :     CG     :  .     DE     :     FH. 

Now,  the  two  triangles  ABC  and  AKB  have  the  common 
base  AB  ;  and  the  triangles  DEF  and  DEM  have  the  common 
case  DE ;  and  since  triangles  on  equal  bases  are  to  each  othei 
as  their  altitudes  (Th.  ix,  Cor.),  we  have 
he  triangle 

ABC     :     ABK     ::     CG     :     AK    or   AB 
and  the  triangle, 

DEF    :     DME     :  :     FH     :     DM   or  DE. 


i02 


vi  E  O  M  E  T  R  y . 


Proportions 

of    I'riangles. 

Bill  wfc  have  proved 

CG     :     AB 

:     FH    :     DE ; 

hence,       ABC     :     ABK 

:  :     DEF    :     Z)iTfJS, 

or,  ahernately, 

. 

ABC     :     DEF 

:     ABK     :     7)3f£. 

But  the  squares  AL  and 
DN  being  each  double  of  the 
triangles  AKB  and  DAfE 
have  the  same  ratio  ;  hence, 

ABC  :  DEF  :  :  AL  :  DN 

K 

THEOREM    XX. 

7^100  similar  polygons  may  he  divided  into  an  equal  numher  of 
triangles,  similar  each  to  each,  and  similarly  placed. 

Let  ABCDE  and  FGHIK  be  two  similar  polycrons. 

From  the  angle  A  draw 
the  diagonals  A-C^  AD :  j) 

and  from-  the;. h-oqiologous 
angle  ^,  draw  .Fi:f,  FL 

.No^<r,  since  •  the'   |toly- 
gons  are  similar,  the  ho- 
mologous angles  B  and  G 
will  be  equal,  and  the  sides  about  the  equal  angles  propor- 
tional (Dof.  1):  that  is, 

AB     :     BC     :  :     FG     :     GIL 

Hence,  tlie  triangles  ABC  and  FGH  have  an  angle  in  each 
equal,  and  the  sides  about  the  equal  angles  proportional  Jiere- 
fore,  they  are  similar  (Th.  xvii),  and  consequently,  the  angle 
ACB  is  equal  to  FHG.  Taking  these  from  the  equal  angles 
HCD  and  GHI,  there  will  remain  ACD  equal  to  FHI.     The 


BOOK     IV.  103 


Proportions    of    Polygoni 


two  triangles  A  CD  and  FHI  will  then  have  an  angle  in  each 
equal,  and  the  sides  about  the  equal  angles  proportional:  hence, 
they  will  be  similar. 

In  the  same  manner  it  may  be  shown  that  the  triangles 
AED  and  FKI  are  similar:  and,  hence,  whatever  bo  the 
number  of  sides  of  the  polygons,  they  may  be  divided  into  an 
equal  number  of  similar  triangles. 

THEOREM     XXI. 

Similar  polygons  arc  to  each  other  as  the  squares  described  on 
their  homologous  sides. 

Let  ABCDE  and  FGNIK^  be  two  similar  polygons  ;  then 
will  they  be  to  each  other 
as  the  squares  described 
on  AD,  FG,  or  any  other 
two  homologous  sides. 

For,  let  ihe  polygons  be 
divided,  as  in  the  last  tlie- 
orem,  into  an  equal   num- 
ber of  similar  triangles.     Then,  by  Tlieorem   XIX,  we   have 
the  triangles 

ABC     :     FGX 

ADC     :     FIN 
ADE     :     FIK 
But  since  the  polygons  are  similar,  the  ratio  of  the  last  ante- 
cedent to  its  consequent,  in   each  of  the  proportions,  is   the 
same :  hence,  we  have  (Bk.  III.  Th.  xiij. 
ABC^-ADC+ADE  :  FGN-\-FIN-\-FIK  :  :  AB"  :  Fff 
that  is,     ABCDE     :     FGNIK     :  :     AB^     :     FG^; 

Hence,  the  areas  of  similar  polygons  are  to  each  other  ae 
the  squares  described  on  their  homoloorous  sides 


A  If     : 

FU' 

DC" 

:     IN' 

DE' 

:    Ik' 

104 


GEOMETRY. 


Proportions     of    Polygons. 


THEOREM     XXII. 


If  similar  polygons  are  inscribed  in  circles,  their  homologous 
sides,  and  also  their  perimeters,  icill  have  the  same  ratio  to  each 
other  as  the  diameters  of  the  circles  in  ivhich  they  are  inscribed. 

Let  ABCDB,  FGNIK, 
be  two  similar  figures,  in- 
scribed in  the  circles  whose 
diameters  are  A  L  and  FM : 
then  will  each  side,  AB, 
BC,  &c.,  of  the  one,  be  to 

the  liomologons  side  FG,  GN,  &c.,  of  the  other,  as  the 
diameter  AL  io  the  diameter  FM.  Also,  the  perimeter 
ABJrBC^  CD  &c.,  will  be  to  the  perimeter  FG  -}-  GN-{.  Nl 
&c.,  as  the  diameter  AL  io  the  diameter  FM. 

For,  draw  the  two  corresponding  diagonals  A  C,  FN^  as  also 
the  lines  BL  and  GM. 

Then,  the  two  triangles  A  CB  and  FNG  will  be  similar 
(Th.  xx)  ;  and  thereforo,  the  angle  A  CB  is  equal  to  the  angle 
FNG.  But,  the  angle  A  CB  is  equal  to  the  angle  ALB,  and 
the  angle  FNG  to  the  angle  FMG  (Bk.  II.  Th.  ix) :  hence, 
the  angle  ALB  is  equal  to  the  angle  FMG  (Ax.  1) ;  and  since 
ABL  and  FGMare  right  angles  (Bk.  II.  Th.  x),  the  two  tri- 
angles ALB  and  FMG  wall  be  equiangular  (Bk.  I.  Th.  xvii. 
Cor.  1),  and  consequently  similar  (Th.  xv). 

Therefore, 

AB     :     FG     ::     AL     :     FM. 

Again,  since  any  two  homologous  sides  are  to  each  other  in 
the  same  ratio  as  AL  to  FM,  we  have  (Bk,  III.  Th.  xii), 
AB-\-BC-{.Cn&c.    :    FG^  GN-\.NI&c.    ::   AL  -.  FM. 


BOOK     IV 


105 


ProportionB    of    Polygons. 


THEOREM     XXIII. 

Similat  polygons  iiiscribed  in  circles  are  to  each  other  as  thi 
squares  of  the  diameters  of  the  circles. 

UiABCDE.FGNJK, 

be  two  polygons  inscribed 
in  the  circles  whose  diam- 
eters are  AL  and  FM: 
then  will  the  polygon 
ABODE,  be  to  the  poly- 
gon FGNIK  as  the  square  of  AL  io  the  square  of  FM. 

For,  the  polygons  being  similar,  are  to  each  other  as  the 
squares  of  their  like  sides  (Th.  xxi) ;  that  is,  as  AB^  to  FG 

But,  by  the  last  theorem, 

AB     '.     FG     ::     AL     :     FM; 
therefore  (Bk  III.  Th.  xiii), 

AB"     '.     FG^     '.:     aT     ',     FM^ ; 
consequently, 

ABODE  :  FGNIK  :  :  aV 
Sch.  If  any  regular  polygon, 
A  BDEFG,  be  inscribed  in  a  circle, 
and  then  the  arcs  AB,  BE,  <fec.,  be 
bisected,  and  lines  be  drawn  through 
these  points  of  bisection,  a  new  poly- 
gon will  be  formed  having  double  the 
number  of  sides.     It  is  plain  that  this  '^  B 

new  polygon  will  differ  less  from  the  circle  than  the  first 
polygon,  and  its  sides  will  lie  nearer  the  circumference  than 
the  sides  '"f  the  first  polygon. 

If  norw,  we  suppose  the  number  of  sides  to  be  conlinuaUy 
increased,  the  length  of  eacli  side  will  constantly  diminish 


FM\ 


t06 


GEOMETRY 


Proportions    of    Circles. 


until  finally  ihe  polygon  will  become 
equal  lo  the  circle,  and  the  perimeter 
will  coincide  with  the  circumference. 
When  this  takes  place,  the  line  CH 
drawn  perpendicular  to  one  of  the 
sides,  will  become  en^ual  to  the  radius 
of  the  circle. 


THEOREM     XXIV. 

The  circumferences  of  circles  arc  to  each  other  as  their  diameters 

Let  there  be  two  circles 
whose  diameters  are  AL 
and  FM:  then  will  their 
circumferences  be  to  each 
other  as  AL  to  FM 

For,  suppose  two  similar  polygons  to  be  inscribed  iii  the 
circles :  their  perimeters  will  be  to  each  other  as  ^L  to  FM 
(Th.  xxii). 

Let  us  now  suppose  the  arcs  which  subtend  the  sides  of  the 
polygons  to  be  bisected,  and  new  polygons  of  double  the  num- 
ber of  sides  to  be  formed :  their  perimeters  will  still  be  to 
each  other  as  AL  to  FM,  and  if  the  number  of  sides  be  in- 
creased until  the  perimeters  coincide  with  the  circumference, 
^'e  shall  have  the  circumferences  to  each  other  as  the  diaro- 
etcis  AL  and  FM. 


THEOREM    XXV. 


77i<?  areas  of  circles  are  to  each  other  as  the  squares  of  theif 
dia^fteters. 


BOOK    IV 


107 


Area    of    the    Circle. 


Let  tlw  le  be  two  circles 
whose  cljaiueters  are  AL 
and  FM :  ilien  will  their 
areas  be  lo  each  other  as 
llie  square  of  AL  to  the 
square  of  FM. 

For  suppose  two  similar  polygons  to  be  inscribed  in  iho 

circles :  then  will  they  be  to  each  other  as  AL  to  FM 
(Th  xxiii). 

Let  us  now  suppose  the  number  of  sides  of  the  polygons  to 
be  increased,  by  bisecting  the  arcs,  until  their  perimeters 
shall  coincide  with  the  circumferences  of  the  circles.  The 
polygons  will  then  become  equal  to  the  circles, and  hence,  the 
areas  of  the  circles  will  be  to  each  oilier  as  the  squares  of  their 
diameters. 

Cot.  Since  the  circumferences  of  circles  arc  to  each  other 
as  their  diameters  (Th.  xxiv),  it  follows,  that  the  areas  which 
are  proportional  to  the  squares  of  the  diameters,  will  also  be 
proportional  to  the  squares  of  the  circumferences 


THEOREM     XXVI. 

The  area  of  a  regular  polygon  inscribed  in  a  circle,  is  equal  to 
half  t/ie  product  of  the  perimeter  and  the  perpendicular  let  fall 
from  the  centre  on  one  of  the  sides. 

Let  C  be  the  centre  of  a  circle  cir- 
cumscribing the  regular  polygon,  and 
CD  a  perpendicular  to  one  of  its  sides  : 
then  will  its  area  be  equal  to  half  the 
product  of  CD  by  the  perimeter. 

For,  from  C  draw  radii  to  the  ver- 
tices of  tlie  angles,  forming  as  many 


108 


GEOMETRY. 


Area  of    Circle 


equal  triangles  as  the  polygon  has 
sides,  in  each  of  which  the  perpen- 
dicular on  the  base  will  be  equal  to 
CD.  Now,  the  area  of  one  of  them, 
as  ACB,  will  be  equal  to  half  the  pro- 
duct of  CD  by  the  base  AB ;  and  the 
same  will  be  true  for  each  of  the  other 
triangles  :  hence,  the  area  of  the  poly- 
gon will  be  equal  to  half  the  product  of  CD  by  the  perimeter 


/y\ 

V 

x/ 

^^--— 

— -^B 

THEOREM     XXVII. 

The  area  of  a  circle  is  equal  to  half  the  product  of  the  radius  b^ 
the  circumference. 

Let  C  be  the  centre  of  a  circle  : 
then  will  its  area  be  equal  to  half  the 
product  of  the  radius  AC  by  the  cir- 
cumference ABE. 

For,  inscribe   within   the    circle   a 
regular  hexagon,  and  draw  CD  perpen- 
dicular  to  one  of   its  sides.      Then, 
the  area  of  the  polygon  will  be  equal  to  half  the  product  o! 
CD  multiplied  by  the  perimeter  (Th.  xxvi). 

Let  us  now  suppose  the  number  of  sides  of  the  polygon  to 
oe  increased,  until  the  perimeter  shall  coincide  with  the  cir- 
cumference ;  the  polygon  will  then  become  equal  to  the  circle, 
and  the  perpendicular  CD  to  the  radius  CA.  Hence,  the  area 
of  the  circle  will  be  equal  to  half  the  product  of  the  radius  >v 
the  circumference. 


0  O  O  K    IV.  109 


Pr  0  olems. 


PROBLEMS 


RELATING    TO    THE   FOURTH    BOOK. 


PROBLEM     I. 

To  divide  a  lijie  into  any  proposed  nujnbcr  of  equal  parti 

Let  AB  be  the  line,  and  let  it  be 
required  to  divide  it  into  four  equal 
parts. 

Draw  any  other  line,  A  C,  forming 
an  angle  with  AB,  and  take  any  dis- 
tance, as  ADj  and  lay  it  off  four  times  on  A  C.  Join  C  and  B 
and  through  the  points  D,  E,  and  F,  draw  parallels  to  CB 
These  parallels  to  BC  will  divide  the  line  AB  into  parts  pro- 
portional to  the  divisions  on  AC  (Th.  xiii) :  that  is,  into  equal 
parts. 

PROBLEM     II. 

To  find  a  third  proportional  to  two  given  tines. 

Let  A  and  B  be  the  given  lines. 

Make  AB  equal  to  A^  and  draw 
ACt  making  an  angle  with  it.  On 
AC  lay  off  ^C  equal  to  5,  and  join 
BC .  then  lay  off  AD,  also  equal  to 
B  and  through  D  draw  DE  parallel  to  BC :  then  will  AE 
be  the  third  proportional  sought. 

For,  since  DE  is  parallel  to  BC,  we  have  (Th.  xiii) 

AB     '.     AC     ::     AD    or     AC     :     AE- 

♦hcrcfore,  AE  is  tlie  third  proportional  sought 
JO 


IK) 


GEOMETRY 


Problems 


PROBLEM    III. 
To  find  a  fourth  proportional  to  the  lines  A,  B,  and  C. 

Place  two  of  the  lines  forming  an 
angle  with  eacli  other  at  A  ;  that  is, 
make  AB  equal  to  A,  and  AC  equal 
B ;  also,  lay  off  AD  equal  to  C. 
Then  join  BC,  aiul  through  D  draw 
DE  parallel  to  BC,  and  AE  will  be  the  fourth  proportional 
sought. 

For.  since  DE  is  parallel  to  BC,  we  have 

AB     :     AC     ::     AD     :     AE; 
therefore,  AE  is  the  fourth  proportional  sought. 


PROBLEM    IV. 

To  find  a  mean  proportional  between  two  given  lines,  A  and  B. 

Make  AB  equal  to  A,  and 
BC  equal  to  B ;  on  AC  de- 
scribe a  semicircle.  Through 
B  draw  BE  perpendicular  to 
4  C,  and  it  will  be  the  mean  proportional  sought  (Th.  xviii.  Cor). 

PROBLEM    V. 

To  make  a  square   which  sJiall  be  equivalent  to  the    mm  of  two 
given  squares. 

Let  A  and  B  be  the  sides  of  the 
given  squares. 

Draw  an  indefinite  line  AB,  and 
make  AB  equal  to  A.     At  B  draw 
BC  perpendicular  to  AB,  and  make 
BC  equal  to  B :  then  draw  AC   and  the  square  described  on 
A  C  will  be  equivalent  to  the  squares  on  A  and  B  (Th.  xii). 


BOOK     IV. 


Ill 


P  r  ob  1  e  ms. 


E 


a 


PROBLEM    VI. 

7V  make  a  square  which  shall  be  equivalent  to  the  differcncJB  be 
tween  two  given  squares. 

l-ct  A  and  B  be  the  sides  of 
<!»e  given  squares. 

I)ravv  an  indefinite  line,  and 
make  CB  equal  to  Ay  and  CD 
equal  to  B.  At  D  draw  DE 
perpendicular  to  CB,  and  with  C  as  a  centre,  and  CB  as  a 
radius,  describe  a  semicircle  meeting  DE  in  E,  and  join  CE: 
then  will  the  square  described  on  ED  be  equal  to  the  difier- 
ence  between  the  given  squares. 

For,  CE  is  equal  to   CB,  that  is,  equal  to  .4,  and   CD  in 
equal  to  B :  and  by  (Th.  xii.  Cor.), 


ED=CE'-CD' 


PROBLEM     VII. 

To  make  a  tnangle  which  shall  be  equivalent  to  a  given  quad' 
rilateral. 

Vi^i  ABCD  be  the  given  quadri- 
ateral. 

Draw  the  diagonal  A  C,  and  through 
D  draw  DjE  parallel  to  u4C,  meeting       ^         ~^  ^ 

BA  produced  at  E.     Join  EC:  then  will   the  triangle  CEB 
be  equi\  alent  to  the  quadrilateral  BD. 

For,  the  two  triangles  ACE  and  ADC,  having  the  same  base 
AC.  and  the  vertices  of  the  angles  D  and  E  in  the  same  line 
DE  parallel  to  AC,  are  equivalent  (Th.  ii).  If  to  each,  we 
add  ACB,  we  shall  then  have  the  triangle  ECB  equivalent  to 
^i^  quadrilateral  BD  (Ax.  2). 


112 


GEOMETRY. 


Problems. 


PROBLEM     VIII. 
To  make  a  triangle  which  shall  be  equivalent  to  a  given  polygon. 

Let  ABCDE  be  the  polygon. 

Draw  the  diagonals  AD,  BD. 
Produce  ^i5  in  both  directions, 
and  through  C  and  E  draw  CG 
and  EF,  respectively  parallel  to 
AD  and  BD :  then  join  FD  and 
DG,  and  the  triangle  FDG  will  be  equivalent  to  the  polygon 
ABCDE, 

For,  the  triangle  AUD  is  equivalent  to  the  triangle  AFD 
and  DBC  to  DBG  (Th.  ii);  and  by  adding  ADB  to  the 
equals,  we  shall  have  the  triangle  FDG  equivalent  to  the 
polygon  ABCDE. 


PROBLEM    IX. 

To  make  a  rectangle  that  shall  be  equivalent  to  a  given  triangle. 

Let  ABC  be  the  given  triangle. 

Bisect  the  base  AB  at  Z),  and  draw 
DH  perpendicular  to  AB.  Through  C, 
the  vertex  of  the  triangle,  draw  CHG 
parallel  to  AB,  and  draw  BG  perpen- 
dicular to  it:  then  will  the  rectangle 
DG  be  equivalent  to  the  triangle  ABC. 

For,  the  triangle  would  be  half  a  rectangle  having  the  fiams 
base  and  altitude  :  hence,  it  is  equivalent  to  J)G^  whose  base 
ifl  the  half  of  AB,  and  altitude  equal  to  that  of  the  triangle. 


BOOK     IV.  113 


Appendix 


PROBLEM    X. 
To  inscribe  a  circle  in  a  regular  polygon. 

Bisect  any  two  sides  of  the  polygon 
by  the  perpendiculars  GO,  FO,  and 
with  their  point  of  intersection  0,  as  a 
centre,  and  OG  as  a  radius  describe 
the  circumference  of  a  circle — this 
circle  will  touch  all  the  sides  of  the 
polygon. 

For,  draw  OA.  Then  in  the  two  right  angled  triangles  OA  G 
and  OAFj  the  side  -4  0  is  common,  and  ^G  is  equal  to  AF, 
since  each  is  half  of  one  of  the  equal  sides  of  the  polygon : 
hence,  OG  is  equal  to  OF(Bk. I.Th.  xix).  In  the  same  man- 
ner it  may  be  shown  that  OH,  OK  and  OL  are  all  equal  to 
each  other  :  hence,  a  circle  described  with  the  centre  O  and 
radius  OF  will  be  inscribed  in  the  polygon. 

C.r.  Hence,  also  the  lines  OA,  ON  &c.,  drawn  to  the 
angles  of  the  polygon  are  equal. 


APPENDIX 

OF     THE     REGULAR     POLYGONS. 

1,  In  a  regular  polygon  the  angles  are  all  equal  to  each 
other  (Def.  3).  If  then,  the  sum  of  the  inward  angles  of  a 
regular  polygon  be  divided  by  the  number  of  angles,  the  quo- 
tient will  be  the  value  of  one  of  the  angles. 

But  the  sum  of  the  inward  angles  is  equal  to  twice  as  many 

right  angles,  wanting  four,  as  the  polygon  has  sides,  and  we 

shall  find  the  value  in  degrees  by  simply  placmg  90^  for  the 

right  angle. 

10* 


114  GEOMETRY. 

Appendix. 

2.  Thus,  for  the  sum  of  all  the  angles  of  an  equilateral 
♦riangle,  we  have 

Gx90°-4x90°  =  540°-3GO°rrl80° 
and  foi  each  angle 

Hence,  each  angle  of  an  equilateral  lrian«^le,  is  equal  to  CO 
degrees. 

3.  For  the  sum  of  all  the  angles  of  a  square,  we  liave 

8  X  90°— 4  X  90°  =  720°  — 360°z:z360% 
and  for  each  of  the  angles 

360°-f-4rr:90° 

4.  For  the  sum  of  all  the  angles  of  a  regular  pentagon,  we 
have 

10X90°— 4x90°  =  900°-360''  =  540'*, 
and  for  each  angle 

540°-^5  =  10S°. 

5.  For  the  sum  of  all  the  angles  of  a  regular  hexagon,  wc 
have 

12x90°-4x90*'=:1080''-3C0°  =  720^ 
and  of  each  angle 

720°^G=:120^ 

6.  For  the  sum  of  the  angles  of  a  regular  heptagon,  we 
have 

14x90''-4x90°=zl2G0°-3G0°z::900^: 
and  for  one  of  the  angles 

900°-f-7-128°  34'-f-. 

7.  For  the  sum  of  the  angles  of  a  regular  octagon,  we  lm^e 

16x90"— 4x90°  =rl440^-360°  =  1080'': 
and  for  each  angle 

1080^^8=  13«'S° 


BOOK       IV 


115 


Regular    Polygons 


8.  Since  the  sum  of  the  angles  about  any  point  is  equal  tc 
four  right  angles  (Bk.  I.  TL  ii.  Cor.  3),  it  may  be  observed  thai 
there  are  only  three  kinds  of  regular  polygons,  which  can  be 
ai  ranged  around  any  point,  as  C,  so  as  exactly  to  fill  up  the 
bpacc.     These  are, 


First. — Six  equilateral  triangles,  in 
which  each  angle  about  C  is  equal  to 
60°,  and  their  sum  to 

G0^XG  =  3G0. 


Second.-  Four  squares,  in  which 
each  ai.glo  Is  equal  to  90°,  and  their 
sum  to 

90°  X  4  =  360° 


c 

Third. — Three  hexagons,  in 
which  each  angle  is  equai  to 
120,  and  the  sum  of  the  three 
to 

120"  X  3=360° 


GEOMETRY. 


BOOK    V. 

OF       PLANES      AND      THEIR      ANGLES. 

DEFINITIONS. 

1 .  A  Straight  line  is  perpendicular  to  a  plane,  when  it  is  per- 
pendicular to  every  straight  line  of  the  plane  which  it  meets. 
The  point  at  which  the  perpendicular  meets  the  plane,  is 
called  iho  foot  of  the  perpendicular. 

2.  If  a  straight  line  is  perpendicular  to  a  plane,  the  piano 
is  also  said  to  be  perpendicular  to  the  line. 

3.  A  line  is  parallel  to  a  plane  when  it  will  not  meet  that 
plane,  to  whatever  distance  both  may  be  produced.  Con* 
versely,  the  plane  is  then  parallel  to  the  line. 

4.  Two  planes  are  parallel  to  each  other,  when  they  will 
not  meet,  to  whatever  distance  both  are  produced. 

5.  If  two  planes  are  not  parallel,  they  intersect  each  other 
in  a  line  that  is  common  to  both  planes :  such  line  is  called 
their  common  intersection. 

6.  The  space  included  between  two  planes  is  called  a 
diedral  angle :  the  planes  are  the  faces  of  the  angle,  and 
their  intersection  the  edge.  A  diedral  angle  is  measured  by 
two  lines,  one  in  each  plane,  and  both  perpendicular  to  the 
common  intersection  at  the  same  point. 

This  angle  may  be  acute,  obtuse,  or  a  right  angle.  When 
it  is  a  j-ight  angle,  the  planes  are  said  to  be  perpendicular  to 
each  other. 


BOOK     V  , 


117 


Of    Planes. 


n 


/ 


B 


I 


I^et  AB  be  a  plane  coinciding  with  ff 

the  I  lane  of  the  paper,  and  ECF  a 
piano  intersecting  it  in  the  line  FII. 
Now,  if  from  any  point  of  the  common 
intersection  as  C,  wc  draw  CD  in  the 
piano  ABj  and  CE  in  the  plane  ECF, 
and  both  perpendicular  to  CF  at  C, 
tlien  will  the  angle  DCE  measure  the  inclination  between 
the  two  planes. 

It  should  be  remembered  that  the  line  EC  is  directly  ovo.i 
the  line  CD. 

7.  A  polyedral  angle  is  the  angular 
space  included  between  several  planes 
meeting  at  the  same  point. 

Thus,  the  polyedral  angle  S  is  formed 
by  the  meeting  of  the  planes  ASB^ 
BSC,   CSD,  DSA. 

8.  The  angle  formed  by  three  planes 
is  called  a  triedral  angle. 

THEOREM    I. 
Two  straight   lines  which   intersect   each  other,  lie  in  the   safM 
plane,  and  determine  its  position. 
Lot  AB  and  ^  C  be  two  straight  lines 
which  intersect  each  other  at  A. 

Through  AB  conceive  a  plane  to  be 
pnascd,  and  let  this  plane  be  turned 
around  AB  until  it  embraces  the  point 
C :  the  plane  will  then  contain  the  two 

Hncs  AB,  AC,  and  if  it  bo   turned  either  way  it  will  depart 
from  the  point  C,  and  consequently  from  the  line  A  C.    Hence, 


118 


GEOMETRY. 


Of    Planei 


the  position  of  the  piano  is  determined 
by  the  single  condition  of  containing 
ilio  twc  straight  lines  AB,  AC. 

Cor.  1.  A  triangle  ABC,  or  three 
points  A,  B,  C,  not  in  a  straight  line, 
determine  the  position  of  a  plane. 

Cor.  2.  Hence,  also,  two  parallels 
AB,  CD  determine  the  position  of  a 
plane.  For  drawing  EF^  we  see  that 
the  plane  of  the  two  straight  lines  AE, 
EF  19  that  of  the  parallels  AB,  CD. 


e/    b 


c   /r 


THEOREM    II. 

.1  perpendicular  ts  the  shortest  line  which  can  he  drawn  from  a 
point  to  a  plane. 

JjCt  A  he  a,  point  above  the  plane 
DE,  and  AB  a.  line  drawn  perpen- 
dicular to  the  plane  :  then  will  AB  be 
shorter  than  any  oblique  line  AC. 

For,  through  B,  the  foot  of  the  per- 
pendicular, draw  BC  to  the  point 
where  the  oblique  line  A  C  meets  the 
plane. 

Now,  since  AB  is  perpendicular  to 
the  plane,  the  angle  ABC  will  be  a 

right  angle  (Def.  1.),  and  consequently  less  than  the  angle  C: 
therefore,  AB,  opposite  the  angle  C,  will  be  less  than  AC 
opposite  the  angle  B  (Bk.  I.  Th.  xi). 


BOOK     V  . 


liO 


Of    Planes 


Cor  It  is  eWdent  that  if  several  lines  be  'Irawn  from  tlie 
point  A  to  the  plane,  that  those  winch  are  nearest  the  perpen- 
dicular AB,  will  be  less  than  those  more  remote. 

Sch.  The  distance  from  a  [)oini  lo  a  plane  is  measured  on 
dio  perpendicular:  hence,  when  thn  </mtfAicc  only  is  named, 
the  shortest  distance  is  always  understood. 


THEOREM    III. 

The  comjnon  intersection  of  two  planes  is  a  straight  line. 

Let  the  two  j)lanes  ABy  CD,  cut 
each  other.  Join  any  two  points  E 
and  F,  in  the  common  intersection, 
by  the  straight  line  EF.  This  line 
will  lie  wholly  in  the  plane  AB,  and 
also  wholly  in  the  plane  CD  (Bk.  I. 
Def.  7) ;  therefore,  it  will  be  in  both 
planes  at  once,  and  consequently,  is 
tlieir  connnon  intersection. 


THEOREM    IV. 
A  straight  line  which  is  perpendicular  to  two  straight  lines  at 
their  point  of  intersection,  will  be  perpendicular  to  the  plane  of 
those  lines. 

Lot  the  line  PA  be  pcri)cn- 
dicular  to  the  two  lines  AD, 
AB:  then  will  it  be  perpendic- 
ular to  the  plane  BC  which  con- 
tains them. 

For,  if  ^P  is  not  perpendicular 
to  the  plane  BC,  suppose  a  plane 


120 


GEOMETRY 


Of    Planes, 


to  be  drawTi  through  A,  tliat  shall 
be  peqiendicular  to  AP 

Now.  every  line  drawn  through 
1,  and  i)erpendicular  to  AP, 
R  ill  be  a  line  of  this  last  plane 
(l)ef.  1):  hence,  this  last  plane 
will  con'cain  the  lines  AB,  AD, 
and  consequently,  a  line  which  is  perpendicular  to  two  linea 
at  the  point  of  intersection,  will  be  perpendicular  to  the  plane 
of  those  lines 


D 

a 

^A 

B 


THEOREM     V. 

If  two  straight  lines  are  perpendicular  to  the  same  plane  they 
will  be  parallel  to  each  other. 

Let  the  two  lines  AB,  CD,  be 
perpendicular  to  the  plane  EF  : 
then  will  they  be  parallel  to  each 
other 

For,  join  B  and  D,  the  points 
in  which  the  lines  meet  the 
plane  EF 

Then,  because  the  lines  AB,  CD,  are  perpendicular  to  the 
plane  ^F,  they  will  be  perpendicular  to  the  line  BD  (DeL  1). 
Now,  if  ^^  and  DC  are  not  parallel,  they  will  meet  at  some 
point  as  0  :  then,  the  triangle  OBD  would  have  two  right 
angles,  which  Ib  impossible  (Bk.  I.  Th.  xvii.  Cor.  4). 

Cor.  If  two  lines  are  parallel^  and  one  of  them  is  perpen- 
diciUar  to  a  plane,  the  other  will  also  be  perpendicular  to  the 
same  plane. 


■' 

'a1 

F 

.5 

/\ 

3           I 

1 

ij  0  0  K     V  . 


121 


Of    Pianos 


THEOREM    VI. 
If  two  planes  intersect  each  other  at  right  angles,  and  a  linn 
ht  drawn  in  one  plane  perpendicular  to  the  common  intersectiont 
this  line  will  be  perpendicular  to  the  other  plane. 

Let  ihc  plane  FE  be  perpen- 
dicular to  MN,  and  ^P  be  drawn 
in  tlic  plane  FE,  and  perpen- 
diculai  to  ilie  common  intersec- 
tion DE:  tJien  will  ^P  be  per- 
pendicular to  the  plane  MN. 

For,  in  the  plane  MN  draw 
*CP  perpendicular  to  the  comn\on 
inicrseciion  DE.  Then,  because  the  planes  MN  and  FE  are 
perpendicular  to  each  other,  the  angle  APC,  which  measures 
their  inclination,  will  be  a  right  angle  (Def.  6).  Therefore, 
the  lino  AP  is  perpendicular  to  the  two  straight  lines  PC  and 
PD ;  hence,  it  is  perpendicular  to  their  plane  MN  (Th.  iv). 

THEOREM    VII. 

tf  one  p^anc  intersect    another  plane,  the  sum  of  the  angles  on 
'he  same  side  will  be  equal  to  two  right  angles. 

Let  the  plane  GEF  intersect 
the  plane  AB  in  the  line  FE : 
then  will  the  sum  of  the  two 
angles  on  the  sKme  side  be  equal 
to  two  right  anjL^les. 

For,  from  any  point,  as  E,  in 

Tud  common  intersection,   draw 

iho  lines  EG  and  DEC,  one  in  each  plane,  and  botn  pcrpen- 

dicrlar  to  the  common  intersection  at  E.     Then,  the  line  GB 

makes,  with  the  line  DEC,  two  angles,  which  fxigether  are 
11 


122 


GEOMETRY. 


Of    Planco 


equal  to  two  right  angles  (Bk  I. 
Th.  ii):  but  these  angles  measure 
the  inclination  of  the  planes  ;  there- 
fore, the  sum  of  the  angles  on  the 
same  side,  which  two  planes  make 
with  each  other,  is  equal  to  two 
right  angles. 

Cnr.  In  like  manner  it  may  be  demonstrated,  that  planes 
which  intersect  each  otlier  have  their  vertical  or  opposite 
angles  equal. 

THEOREM     VIII. 

Two  planes  whch  are  perpendicular  to  the  same  straight  line  out 
parallel  to  each  other. 

Let  the  planes  MN  and  PQ 
be  perpendicular  to  the  line  AB:  q 
then  will  they  be  parallel.  "^^ 

For,  if  they  can  meet  any 
where,  let  O  be  one  of  their 
their  common  points,  and  draw 
OB,  in  the  plane  PQ,  and  OA, 
in  the  plane  MN.  ^ 

Now,  since  AB  is  perpendicular  to  both  planes,  it  will 
be  perpendicular  to  OB  and  OA  (Def.  1):  hence,  the  triangle 
GAB  will  have  two  right  angles,  which  is  impossible  (Bk.  1. 
Th.  xvii.  Cor.  4) ;  therefore,  the  pianes  can  have  no  point,  r.a 
O,  in  common,  and  consequently,  they  are  parallel  (Def  'J). 


-^ 

p\ 

N 

\  \ 

\ 

\  ' 

•    ^ 

THEOREM    IX. 


If  a  plane  cuts  two  parallel  planes,  the  lines  of  intersection  wiU 
he  parallel 


COOK     V  . 


12a 


Of    Planes. 


Let  ikc  parallel  planes  Mi\  and 
PA  be  intursi'ccted  by  the  plane 
EH :  then  will  the  lines  of  inter- 
section EF,  GHy  bo  parallel. 

For,  if  tlie  lines  EF,  GH,  were 
not  p'irallel,  they  would  meet  each 
other  if  sufficiently  produced,  since 
they  lie  in  the  same  plane.  If  this 
were  so, the  planes  MN,  P^,  would 
meet  each  other,  and,  consequently,  could  not  be  parallel; 
which  would  be  contrary  to  the  supposition. 


/^ 


E 

G, 


THEOREM     X. 

Ij  two  lines  are  parallel  to  a  third  line,  (hough  not  in  the  sanu 
plane  with  ity  they  luill  be  parallel  to  carh  other. 

Let  the  lines  AB  and  CD  be  each 
parallel  to  tlie  third  line  EF,  though 
not  in  the  same  plane  with  it :  then 
vnW  they  be  parallel  to  each  other. 

Foi  since  ilF  and  CD  are  parallel, 
tliey  Avill  lie  in  the  same  plane  FC 
(Th.  i.  Cor.  2),  and  AD,  EF  will  also 
lie  in  the  plane  EB. 

At  any  point,  G,  in  the  line  EF,  let  GI  and  GH  be  drawn 
in  the  planes  FC,  BE,  and  each  perpendicular  to  FE  at  G 

Then,  since  the  line  EF  is  perpendicular  to  the  lines  Gil 
Gl,  it  will  be  perpendicular  to  the  plane  HGI  (Th.  iv).  And 
since  FE  is  perpendicular  to  the  plane  HGI,  its  parallels 
AB  and  DC  will  also  be  perpendicular  to  the  same  plane 
(Th.  v).  Hence,  since  the  two  lines  AB,  CD,  arc  both  per* 
pendicular  to  the  plane  HGL  the>  will  be  parallel  to  each  other 


B 


D 


124 


GEOMETRY 


Of    Planes 


TIIEOREiM     XI. 

If  two  angles^  not  situated  in  the  same  plane^  have  their  stdes 
paraUel  and  lying  in  the  same  direction,  the  angles  will  li 
equal. 

Lot  tlie  angles  ACE  and  BDF 
have  tlin  sides  A  C  parallel  to  BD, 
and  CE  to  DF :  then  will  the  angle 
ACE  be  equal  to  the  angle  BDF. 

For,  make  AC  equal  to  BD,  and 
CE  equal  to  DF,  and  join  AB,  CD, 
and  EF ;  also,  draw  AE,  BF. 

Now  since  A  C  is  equal  and  par- 
allel to  BD,  the  figure  AD  will  be  a 
parallelogram  (Bk.  1.  Th.  xxv);  there- 
fore, AB  is  equal  and  parallel  to  CD. 

Again,  since  CE  is  equal  and  parallel  to  DF,  CF  will  be 
a  parallelogram,  and  EF  will  be  equal  and  parallel  to  CD. 
Then,  since  A  B  and  EF  are  both  parallel  to  CD,  they  will 
be  parallel  to  each  other  (Th.  x) ;  and  since  they  are  each 
equal  to  CD,  they  will  be  equal  to  each  other.  Hence,  the 
figure  BAEF  is  a  parallelogram  (Bk.  I.  Th.  xxv),  and  conse- 
quently, AE  is  equal  to  BF.  Hence,  the  two  triangles  ACE 
and  BDF  have  the  three  sides  of  the  one  equal  to  the  throe 
sides  of  the  other,  each  to  each,  and  therefore  the  angle  ACJl 
is  equal  to  the  angle  BDF  (Bk.  I.  Th.  viii). 


THEOREM     XII. 


If  two  planes  are  parallel,  a  straight  line  which  is  perpendicular 
to  the  one  will  also  be  perpendicular  to  the  other. 


BOOK     V 


I2d 


Of    Planes. 


{ 


A'. 


\ 


Let  MN  and  PQ  be  two  par- 
allel planes,  and  let  AB  be  per- 
pendicular to  MN :  then  will  it 
be  perpendicular  to  PQ. 

For,  draw  any  line,  BC,  in  tlie 
plane  PQ^  and  through  the  lines 
AB,  BC,  suppose  the  plane 
ABC  to  be  drawn,  intersecting  ^ 

the  plane  MN  in  the  line  AD  :  then,  the  intersection  AD  will 
be  parallel  to  BC  (Th.  ix).  But  since  AB  is  perpendicular 
to  the  plane  iO/",  it  will  be  perpendicular  to  the  straight  hne 
AD^  and  consequently,  to  its  parallel  BC  (Bk.  I.  Th.  xii.  Cor.) 

In  like  manner,  AB  might  be  proved  perpendicular  to  any 
other  line  of  the  plane  PQ,  which  should  pass  through  B ; 
hence,  it  is  perpendicular  to  the  plane  (Def.  1). 

Cor.  It  from  any  point  as  H^ 
any  oblique  lines,  as  IIEF,  HDC, 
be  drawn,  the  parallel  planes  will 
cut  these  lines  proportionally. 

For,  draw  HAB  perpendicular 
to  the  plane  MN :  then,  by  the 
theorem,  it  will  also  be  perpendi- 
cular to  P^.  Then  draw  AD,  AE, 
BC,  BF.  Now,  since  AE,  BF, 
\re  tlie  intersections  of  the  plane 
FliB,  with  the  two  parallel  planes  MN,  PQ,  they  are  parai- 
k'l  (Th  ix.) ;  and  so  also  are  AD,  BC. 

Then,         HA     :     HB      :      HE    :    HF, 
and  HA     \     HB      '.      HD     :     EC, 

hence,        HE    :     HF    .  :     HD     :    HC 


GEOMETRY. 


BOOK     VI. 


OF      SOLIDS. 


DEFIXITIOXS 


1.  Evcrj^  solid  bounded  by  planes  is  called  a. polycdron. 

2.  The  planes  which  bound  a  polyedron  are  called  faces. 
The  straight  lines  in  which  the  faces  intersect  each  other, 
are  called  the  edges  of  the  polyedron,  and  the  points  at  which 
the  edges  intersect,  are  called  the  vertices  of  the  angles,  or 
w^ertices  of  the  polyedron. 

3.  Two  polyedrons  are  similar,  when  they  are  contained  by 

the  same  number  of  similar  planes,  and  have  their  polyedral 
angles  equal,  each  to  each. 

4.  A  prism  is  a  solid,  whose  ends 
arc  equal  polygons,  and  whose  side 
faces  are  parallelograms. 

Thus,  the  prism  whose  lower  base 
is  ilie  pentagon  ABODE,  terminates 
in  an  equal  and  parallel  pentagon 
FGHIK,  which  is  called  the  vpper 
base.  The  side  faces  of  the  prism 
are  the  parallelograms  DH,  DK,  EF, 
AG,  and  BH.  These  are  called  the  convex, or  /a^era/  surface 
of  th(x  ori  3m 


BOOK     V 


127 


Of    the    Prism 


5.  The  aliitiidc  of  a  prism  is  the  distance  between  its  upper 
and  lower  bases  :  tliat  is,  it  is  a  line  drawn  from  a  point  of  iho 
upper  base,  perpendicular,  to  the  lower  base 


6,  A  right  prism  is  one  in  which 
the  edges  AF,  BG,  EK,  HC,  and 
D/,  are  perpendicular  to  the  bases. 
In  the  right  prism,  either  of  the  per- 
pendicular edges  is  equal  to  the 
altitude.  In  the  oblique  prism  the 
altitude  is  less  than  the  edge. 


F 

\ 

: 
i 

i 

y] 

1 

A 

G 

\ 

7 

D 

1 

i         C 

7.  A  prism  whose  base  is  a  triangle,  is  called  a  triangular 
prism  ;  if  the  base  is  a  quadrangle,  ii  is  called  a  quadrangidai 
prism ;  if  a  pentagon,  a  pcntaj][onal  prism ;  if  a  hexagon  a 
hexagonal  jirism ;  (Vc. 


8  A  prism  whose  base  is  a  paralielo- 
graut,  ami  all  of  whose  faces  are  also 
parallelojrrams,  is  called  a  parallelopipe- 
don.  If  all  the  faces  are  rectangles,  it  is 
called  a  rectangular  parallelopipedon. 


4 


::^ 


9,  If  the  faces  ol  the  rectangidar  par- 
allelopipedon are  squares,  the  solid  is 
called  a  cube:  hence,  the  cube  is  a  prism 
bounded  by  bIx  equal  squares 


128 


G  E  O  IVI  E  T  R  Y 


Of    the    Pyramid 


10.  A  pyramid  is  a  solid,  formed  by 
several  triangles  united  at  the  same 
point  S,  and  terminating  in  the  difTcr- 
ent  sides  of  a  polygon  ABODE. 

The  polygon  ABODE ^  is  called  the 
base  of  the  pyramid ;  the  point  *S,  is 
called  the  vertex,  and  the  triangles 
ASB,  BSO,  OSD,  DSE,  and  ESA, 
form  its  lateral,  or  convex  surface. 


1 1 .  A  pyramid  Avhose  base  is  a  triangle,  is  caOed  a  tnun^ 
gular  pyramid;  if  the  base  is  a  quadrangle,  it  is  called  a 
quadrangular  p}Tamid ;  if  a  pentagon,  it  is  called  a  petagonaj 
pjramid;  if  the  base  is  a  hexagon,  it  is  called  a  hexagonal 
l>>Tamid;  &c. 


12.  The  altitude  of  a  pyramid,  is  the 
perpendicular  let  fall  from  the  vertex, 
upon  the  plane  of  the  base.  Thus, 
SO  is  the  altitude  of  the  pyramid 
^-^ABODE. 


13.  When  the  base  of  a  pjTamid  is  a  regular  polygon,  a;ul 
the  perpendicular  SO  passes  through  the  middle  point  of  the 
base,  the  pyramid  is  called  a  right  pyramid,  and  the  line 
SO  is  called  the  axis 


BOOK     VI. 


]29 


Pyramid    and    Cylinder 


14.  The  slant  height  of  a  right 
pyramid,  is  a  line  drawn  from  the  ver- 
tex, perpendicular  to  one  of  the  sides 
of  the  polygon  which  forms  its  base. 
Thus.  SF  is  the  slant  heigiit  of  the 
P}Tamid  S'-ABCDE. 


15.  If  from  the  pyramid  S—ABCDE 
the  pyramid  S — abcde  be  cut  off  by  a 
plane  parallel  to  the  base,  the  remain- 
mg  solid,  below  the  plane,  is  called 
the  frustum  of  a  pyramid. 

The  altitude  of  a  frustum  is  the  per- 
pendicular distance  between  the  upper 
and  lower  planes. 


16.  A  Cylinder  is  a  sold,  described  by 
the  revolution  of  a  rectangle,  AEFD, 
about  a  fixed  side,  EF. 

As  the  rectangle  AEFD,  turns  around 
ihe  side  EF,  like  a  door  upon  its  hinges, 
the  lines  AE  and  FD  describe  circles, 
and  the  line  AD  describes  the  convex  sur- 
face of  the  cylinder. 

The  circle  described  by  the  line  A  E,  is  called  the  lowei 
base  of  the  cylinder,  and  the  circle  described  by  Z)F,  is  called 
vhc  upper  base. 


vso 


GEOMETRY 


Of    the    Cylinder 


The  immovable  line  EF  is  called  the  axis  3f  the  cylinder 
A  cylinder,  therefore,  is  a  round  body  with  circular  ends 


17,  If  a  plane  be  passed  through  the 
axis  of  a  cylinder,  it  will  intersect  the  cylin- 
der in  a  rectangle,  FG,  which  is  double 
the  revolving  rectangle  DF!. 


18.  If  a  cylinder  be  cut  by  a  plane  par- 
allel to  the  base,  the  section  will  be  a  cir- 
cle equal  to  the  base.  For,  while  the 
Bide  FC,  of  the  rectangle  MC,  describes 
the  lower  base,  the  equal  side  MPj  will 
describe  the  circle  MLKN,  equal  to  the 
lower  base 


19  If  a  polygon  be  inscribed  in  the 
lower  base  of  a  cylinder,  and  a  corres- 
ponding polygon  be  inscribed  in  the  upper 
base,  and  their  vertices  be  joined  by 
straight  lines,  the  prism  thus  formed  is 
said  to  be  inscribed  in  the  cylinder. 


BOOK      VI. 


131 


Of    the    Cone 


20.  A  cone  is  a  solid,  described  by 
the  revolution  of  a  right  angled  triangle, 
ABC,  about  one  of  its  sides,  CD 

The  circle  described  by  the  revolving 
side,  AB^  is  called  the  base  of  tlie  cone. 

The  hypothenuse,  AC^  is  called  the 
slant  height  of  the  cone,  and  the  surface 
described  by  it,  is  called  the  convex 
surface  of  the  cone. 

The  side  of  the  triangle,  C5,  which  remains  fixed,  is  called 
the  axisy  or  altitude  of  the  cone,  and  the  point  C,  the  vertex 
of  the  cone. 

21.  If  a  cone  be  cut  by  a  plane  par- 
allel to  the  base,  the  section  will  be  a 
circle.  For,  while  in  the  revolution  of 
the  right  angled  triangle  SAC,  the  line 
CA  describes  the  base  of  the  cone,  its 
parallel  FG  will  describe  a  circle 
FKHI,  parallel  to  the  base.  If  from 
the  cone  S—CDB,ihe  cone  S—FKH 
be  taken  away,  the  remaining  part  is 
called  the  frustum  of  the  cone 


22.  If  a  polygon  be  inscribed 
in  the  base  of  a  cone,  and  straight 
lines  be  drawn  from  its  vertices 
to  the  vertex  of  the  cone,  the  pyra- 
mid thus  formed  is  said  to  be  in- 
scribed in  the  cone.  Thus,  the 
pjTamid  S — ABCD  is  inscribed  in 
the  cono 


132 


G  E  O  ]\1  E  T  Jl  Y 


Of    the    Sphere, 


23.  Two  cylinders  are  similar,  when  the  iiameters  of  theii 
ba^cs  are  proportional  to  their  altitudes. 

'M.  Two  cones  are  also  similar,  when  the  diameters  of  tlieii 
bases  are  proportional  to  their  altitudes. 

25.  A  sphere  is  a  solid  terminated  by  a  curved  surface,  all 
tlie  points  of  which  are  equally  distant  from  a  certain  poml 
within  called  the  centre. 


26.  The  sphere  may  be  described 
by  revolving  a  semicircle,  ABD, 
about  the  diameter  AD.  The  plane 
will  describe  the  solid  sphere,  and 
the  semicircumference  ABD  will 
describe  the  surface. 


27.  The  radius  of  a  sphere  is  a 
line  drawn  from  the  centre  to  any 
point  of  the  circumference.  Thus, 
CA  is  a  radius. 


28.  The  diameter  of  a  sphere  is 
a  lino  passing  through  the  centre, 
end  terminated  by  the  circumfer 
ence.     Thus.  AD  is  a  diameter 


BOOK     VI 


133 


Of    the    Sphere 


29.  All  diameters  of  a  sphere  are  equal  to  each  other ;  and 
each  is  double  a  radius. 

30.  The  axis  of  a  sphere  is  any  line  about  which  it  ro- 
rolves ;  and  the  points  at  which  the  axis  meets  the  surface, 
are  called  the  poles. 


31.  A  plane  is  tangent  to  a  sphere 
when  it  has  but  one  point  in  com-         ^^ 
mon  with  it.     Thus,  ^i3  is  a  tan- 
gent plane,  touching  the  sphere  at  B. 


32.  A  zone  is  a  portion  of  the  sur- 
face of  a  sphere,  included  between 
two  parallel  planes  which  form  its 
bases.  Thus,  the  part  of  the  surface 
included  between  the  planes  AE 
and  DF  is  a  zone.  The  bases  of 
this  zone  are  the  two  circles  whose 
diameters  are  AE  and  DF. 


33.  One  of  the  planes  which 
bound  a  zone  may  become  tangent 
to  the  sphere ;  in  which  case  the 
zone  will  have  but  one  base.  Thus, 
if  one  plane  be  tangent  to  the  sphere 
at  A,  and  another  plane  cut  it  in  the 
circle  DF,  the  zone  included  be- 
tween them,  will  have  but  one  base. 
12 


134 


G  E  0  M  E  T  li  Y 


Of    the     P I  ;  s  in  . 


34.  A  spherical  segment  is  a  portion  of  the  solid  sphere  in- 
chided  between  two  parallel  planes.  These  parallel  planes 
arc  its  bases.  If  one  of  the  planes  is  tangent  to  the  sphere, 
the  segment  will  have  but  one  base. 

35.  The  altitude  of  a  zone  or  segment,  is  the  distance  be 
!i\-een  the  parallel  planes  which  form  its  bases 


THEOREM    I. 

Tlic  convex  surface  of  a  right  prism  is  equal  to  (he  perimeter  of 
%ts  base  ?nultiplicd  Inj  its  altitude. 

Let   ABODE— K    be     a    right 
prism:  then  will  its  convex  surface 
be  equal  to 
{AB^  BC+CD+DE  +  EA)xAF. 

For,  the  convex  surface  is  equal 
to  the  sum  of  the  rectangles  AG^ 
BH,  CI,  DK,  and  EF,  which  com- 
pose it ;  and  the  area  of  each  rectan- 
gle is  equal  to  the  product  of  its  base 
by  its  altitude.  But  the  altitude  of  each  rectangle  is  equal  to 
the  altitude  of  the  prism  :  hence,  their  areas,  that  is,  the  con- 
vex surftice  of  the  prism,  is  equal  to 

{AB-\-BC-\-CD-hDE-\-EA)xAF; 

that  is,  equal  to  the  perimeter  of  the  base  of  the  prism  nmhi 
plied  by  its  altitude. 


THEOREM    II. 

T%€  convex  surface  of  a  cylinder  is  equal  to  the  circumference  oj 
its  base  multiplied  by  its  altit'ude 


BOOK     VI. 


135 


Of    the    Prism 


Let  DB  be  a  cylinder,  and  AB  the 
diameter  of  its  base  :  the  convex  sur- 
face will  then  be  equal  to  the  altitude 
AD  mullij  lied  by  the  circumference 
of  the  base. 

For,  suppose  a  regular  prism  to  be 
insc/ibed  within  the  cylinder.  Then, 
the  convex  surface  of  the  prism  will  be 
equal  to  the  perimeter  of  the  base  mul- 
tiplied by  the  altitude  (Th.  i).  But  the  altitude  of  the  prism 
is  the  same  as  that  of  the  cylinder ;  and  if  we  suppose  the 
sides  of  the  polygon,  which  forms  the  base  of  the  prism,  to 
be  indefinitely  increased,  the  polygon  will  become  the  circle 
(Bk.  IV.Th.  xxiii.  Sch.),  in  which  case,  its  perimeter  will  become 
the  circumference,  and  the  prism  will  coincide  with  the  cyhnder. 
But  its  convex  surfjice  is  still  equal  to  the  perimeter  of  its  base 
multiphed  by  its  altitude  :  hence,  the  convex  surface  of  a  cylin- 
der is  equal  to  the  circumference  of  its  base  nmltiplied  by  its  al- 
titude. 

THEOREM    III. 

In  every  prism  the  sections  formed  by  planes  parallel  to  the  bast 
are  equal  polygons. 

Let  AG  he  any  prism,  and  IL  a  sec- 
lion  made  by  a  plane  parallel  to  the 
base  AC:  then  will  the  polygon  IL 
be  equal  to  A  C, 

For,  the  two  planes  AC,  IL,  being 
parallel,  the  lines  AB,  IK,  in  which 
they  intersect  the  plane  AF,  will  also 
be  parallel  (Bk.  V.  Th.  ix).  For  a 
like  reason,  BC  and  KL  will  be  par- 


136 


GEOMETRY 


Of    the    Pyramid. 


M 


K 


allcl;  also,  CD  will  be  parallel  to  LM, 
and  AD  to  IM. 

But,  since  AI  and  BK  are  parallel, 
the  figure  AK  is  a  parallelogram  : 
hence  AB  is  equal  to  IK  (Bk.  I. 
Th.  xxiii).  In  the  same  way  it  may  be 
shown  that  BC  is  equal  to  KL,  CD  to 
LM,  and  AD  to  IM. 

But,  since  the  sides  of  the  polygon 
AC  are  respectively  parallel  to  the 
sides  of  the  polygon  IL,  it  follows  that  their  corresponding 
angles  are  equal  (Bk.  V.Th.  xi),  viz.,  the  angle  A  to  the  angle 
/,  the  angle  B  to  K,  the  angle  C  to  L,  and  the  angle  M  to  D ; 
hence,  the  polygon  IL  is  equal  to  A  C. 

Sch.  It  was  shown  in  Definition  18,  that  the  section  of  a 
cylinder,  by  a  plane  parallel  to  the  base,  is  a  circle  equal  to 
the  base. 


THEOREM    IV. 

If  a  pyramid  be  cut  by  a  plane  parallel  to  the  base, 

I.  The  edges  and  altitude  vnll  be  divided  proportionally. 

II.  The  section  will  be  a  polygon  similar  to  the  base. 
Let  the    pyramid    S—ABCDE,   of 

which  SO  is  the  altitude,  be  cut  by  the 
plane  abcde  parallel  to  the  base :  then 
will, 

Sa     :     SA     :  :     Sb     :     SB, 

and  the  same  for  the  other  edges ;  and 
the  polygon  abcde  will  be  s  jnilar  to  the 
base  ABCDE. 

First.  Since  the  planes  ABC  and  abr 


B  O  O  R    V  1  .  137 


Of    the    Pyramid 


we  parallel,  their  intersections,  AB,  ab,  by  the  plane  SAB^ 
will  also  be  parallel  (Bk.  V.  Th.  ix) ;  hence,  the  trVangloa 
SABj  sab,  are  similar,  and  we  have 

SJ  Sa     :  :     SB     :     Sb ; 

for  a  similar  reason,  mc  have 

SB     :     Sb     :        SC     :     Sc; 

end  the  same  for  the  other  edges  •  hence,  the  edges  SA,  SB, 
SC,  &c.,  are  cut  proportionally  at  the  points  a,  b,  c,  <fec. 
The  altitude  SO  is  likewise  cut  proportionally  at  the  point 
The  altitude  *S0  is  likewise  cut  in  the  same  proportion  at 
the  point  o  ;  for,  since  ^  0  is  parallel  to  bo,  we  have 

SO     :     So     :  :     SB     :     Sb. 

Secondly.  Since  ab  is  parallel  to  AB,  be  to  BC,  cd  to  CD 
6lc.  ;  tlie  angle  abc  is  equal  to  ^SC,  the  angle  bed  to  BCD 
and  80  on  (Bk.  V.  Th.  xi). 

Also,  by  reason  of  the  similar  triangles,  SAB,  Sab,  we  have 

AB     :     ab     :       SB     :     Sb, 

and  by  reason  of  the  similar  triangles  SBC,  Sbcj  we  have 
SB     :     Sb     :  :     BC     :     be; 

hence  (Bk    III.  Th.  v), 

AB     :     ab     :  :     BC     :     be; 

nnd  for  a  similar  reason,  we  also  have 

BC     :     be     :  :      CD     :     cd,  <fec. 

Ikucc,  tne  polygons  ABCDE,  abcde,  having  their  angles 
respectively  equal,  and  their  homologous  sides  [roporlional 
are  similar. 


ms 


0  E  O  M  £  1'  R  Y 


Of    the    Pyramid 


THEOREM     V. 
If  two  pyramids^  having  equal  altitudes  and  their  bases  in  the 
same  plane,  he  intersected  hy  planes  par  illel  to  the  plane  of  iha 
bases,  the  sections  in  each  pyramid  v:ill  /f  proportional  to  the  bases 

Let  S— ABODE,  and 
S — XYZ,  be  two  pyra- 
mids, having  a  common 
rertox,  and  their  bases  sit- 
uated in  the  same  plane. 
If  these  pyramids  are  cut 
by  a  plane  parallel  to  the 
plane  of  their  bases,  giv- 
ing the  sections  abcde, 
xyz,  then  will  the  sections 
abcde,  xyz,  be  to  each  other  as  the  bases  ABCDE,  XYZ. 

For,  the  polygons  ABCDE,  abcde,  being  similar,  theii  sur- 
faces arc  as  the  squares  of  the  homologous  sides  AB,  ab ; 

but  AB     :     ab     :  :     SA     :     Sa ; 

hence,        ABCDE     :     abode     :  :      SA^     :      Sa 
For  the  same  reason, 

XYZ     :     X7JZ     ::     SX^     :     Si\ 
But  since  abc  and  xyz  are  in  one  plane,  the  lines  SA,  Sa,  SX, 
Sx,  are  proportional  to  SO,  So  :  (Bk.  V.  Th  xii.  Cor.),  therefore, 

SA     :     Sa     :  :     SX     :     Sx  : 
hence,      ABCDE     :     abcde     :  :     XYZ     :     xyz. 
consequently,  the  sections  abcde,  xyz,  are  to  each  other  as  the 
bases  ABCDE,  XYZ. 

Cor.  If  the  bases  ABCDE,  XYZ,  are  equivalent,  any  sec- 
lions  abcde,  xyz,  made  at  equal  distances  from  the  bases,  will 
be  also  equivalent 


BOOK      VI 


139 


Of    the    l*yramid. 


TIIEOllEM    VI. 

The  convex  surface  of  a   right  pyramid  is  equal  to  half  th^  prt"^ 
duct  cf  the  perimeter  of  its  base  multiplied  by  the  slant  height. 

Lai  S— A  BCD E  be  a  right  ppa- 
mid,  SF  its  slant  height:  then  will  its 
convex  surface  be  equal  to  half  the 
product 

SFx(AB^BC-\-CD-\-DE-\-EA). 

For,  since  ilie  pyramid  is  right,  the 
point  0,  in  which  the  axis  meets  the 
base,  is  the  centre  of  the  polyi^on 
ABODE;  hence,  the  lines  OA,  OB, 
&,c  drawn  to  the  vertices  of  the  base, 
arc  equal  (Bk.  IV.  prob.  x.Cor).  t 

Now,  in  the  right  angled  triangles  SAO,  SBO,  the  bases 
and  perpendiculars  are  equal :  hence,  the  hypothenuses  are 
equal ;  and  in  the  same  way  it  may  be  proved  that  all  the 
edges  of  the  pyTamid  are  equal.  The  triangles,  therefore, 
which  form  the  convex  surface  of  the  prism,  arc  all  equal  tn 
each  other. 

But  the  area  of  either  of  these  triangles,  as  SAB,  is  equal 
to  half  the  product  of  the  base  AB,  by  the  slant  height  of  the 
pyramid  SF:  hence,  the  area  of  all  the  triangles,  which  form 
the  convex  surfiice  of  the  pyramid,  is  equal  to  half  the  proJunt 
of  the  perimeter  of  the  base  by  the  slant  height. 


TIIEORE.M    VII. 


The  convex  surface  of  the  frustum  of  a  regular  pyramid  is 
t<pial  to  half  the  sum  of  the  perimeters  of  the  upper  and  iowef 
bases  multjnlied  by  the  slant  height. 


140 


GEOMETRY. 


Of    the    Cone. 


Let  a — ABODE  be  the  frusUim  of  a 
regular  pyramid :  then  will  its  convex 
surface  be  equal  to  half  the  product  of 
the  perimeter  of  its  two  bases  multi- 
plied by  the  slant  height  Ff. 

For,  since  the  upper  base  abcde,  is 
similar  to    the   lower    base   ABCDE 
(Th.  iv),  and  since  ABCDE  is  a  regulai  polygon,  it  follows 
that  the  sides  ah,  be,  cd,  de,  and  ea,  are  all  equal  to  each  other. 

Hence,  the  trapezoids  EAae,  ABba,  &c.,  which  form  the 
convex  surface  of  the  frustum  are  equal.  But  the  perpen- 
dicular distance  between  the  parallel  sides  of  these  trapezoids 
is  equal  to  Ef,  the  slant  height  of  the  frustum. 

Now,  the  area  of  either  of  the  trapezoids,  as  AEea,  is  equal 
to  half  the  product  o(  Ffx(EA-]-ea)  (Bk.  IV.  Th.  x):  hence, 
the  area  of  all  of  t?iem,  that  is,  the  convex  surface  of  the 
frustum,  is  equal  to  half  the  sum  of  the  perimeters  of  the 
upper  and  lower  bases,  multiplied  by  the  slant  height. 


THEOREM     VIII. 

The  convex  surface  of  a  cone  is  equal  to  half  the  product  of  th* 
circumference  of  the  base  multiplied  by  the  slant  height. 

In  the  circle  which  forms  the  base 
of  the  cone,  inscribe  a  regular  poly- 
gon, and  join  the  vertices  with  the 
vertex  S,  of  the  cone  We  shall 
then  have  a  right  pyramid  in- 
scribed in  the  cone. 

The  convex  surface  of  this  pyra- 
mid will  be  eoual  to  half  the  product 


BOOK      VI 


141 


Of    the    Cone 


of  the  perimeter  of  the  base  by  the 
slant  height  (Th.  vi). 

Let  us  now  suppose  the  number 
of  sides  of  the  polygon  to  be  indefi- 
nitely increased :  the  polygon  will 
then  coincide  with  the  base  of  the 
cone,  the  pyramid  will  become  the 
cone,  and  the  line  Sf  which  meas- 
ures the  slant  height  of  the  pyramid, 
will  then  measure  the  slant  height 
of  the  cone. 

Hence,  the  convex  surface  of  the  cone  is  equal  to  half  the 
product  of  the  slant  height  by  tlie  circumference  of  the  base. 


THEOREM    IX. 

The  convex  surface  of  the  frustum  of  a  cone  is  equal  to  half 
the  su?n  of  the  circumferences  of  its  two  bases  multiplied  by  tht 
nlant  height. 

For,  if  we  suppose  the  frustum  of 
a  right  pyramid  to  be  inscribed  in 
the  frustum  of  a  cone,  its  convex 
surface  will  be  equal  to  half  the  pro- 
duct of  its  slant  height  by  the  perim- 
eters of  its  two  bases.  But  if  we 
increase  the  number  of  sides  of  the 

polygon  indefinitely,  the  frustum  of  the  pyramid  will  become 
the  frustum  of  the  cone  :  hence,  the  area  of  the  frustum  of  the 
cone  i3  equal  to  half  the  sum  of  the  circumferences  of  its  two 
btses  multiplied  by  the  slant  height 


i42 


G  E  O  ]M  E  T  R  Y  . 


Of    Parallelopipedons. 


THEOREM     X. 

Two  rectangular  parallelopipedons^  having  equal   altitudes  and. 
equal  bases,  are  equal. 

Let  E — ABCD,  and  F — KG  HI,  be  uvo  rectangular  par 
allolopipeJons    having   equal 
bases,  AC  and  KH,  and  equal 
altitudes,  AE  and  KF :  then 
will  lliey  be  equal. 

For,  apply  the  base  of  the 
one   parallelopipedon  to  that 


E 

\ 
\ 

\\ 

\ 

K 

'\ 

4       ^          ^ 

<. 

\ 

N 

B  C         G  H 

of  the  other,  and  since  the  bases  are  equal,  they  will  coincide 
Again,  since  the  edges  are  perpendicular  to  the  bases,  the 
edges  of.  the  one  parallelopipedon  will  coincide  with  those  of 
the  other;  and  since  the  altitude  AE  is  equal  to  KF,  ihe 
planes  of  the  upper  bases  will  coincide.  Hence,  the  paral- 
lelopipedons will  coincide,  and  consequently  they  are  equal 


THEOREM     XI. 

Two  rectangular  parallelopipedons,  which  have  the  same  base,  are 

to  each  other  as  their  altitudes. 


Let  the  parallelopipedons  AG,  Ah, 
have  the  same  base  BD,  then  will  they 
be  to  each  other  as  their  altitudes  AE 
AL 

Suppose  the  altitudes  AE,  AT,  to 
be  to  each  other  as  two  whole  num- 
bers, as  15  is  to  8,  for  example.  Di- 
vide AE  into  15  equal  parts,  whereof 
Al  will  contain  8 ;  and  through  x.  y,  z, 
&c..  the  points  of  division,  draw  planes 


E 


y 

x-Y 
A 


T 


K 


H 


M 


\ 


f 


BOOK     VI 


143 


Of    Parallolopipodons 


parallel  to  the  base.  These  planes 
will  cut  the  solid  AG  into  15  partial 
parallclopipcdons,  all  equal  to  each 
Otlicr,  because  they  liavo  equal  bases 
ftinl  equal  altitudes — equal  bases,  since 
cveiy  section,  IL,  made  parallel  to 
ihc  base  BDj  of  a  prism,  is  equal 
to  that  base  ;  equal  altitudes,  because 
the  altitudes  are  the  equal  divisions  Ax^ 
xy,yz,  &c.  Hut  of  these  15  equal  par- 
allelopipedons,  8  are  contained  in  AL; 
hence,  solid  AG  :  solid  AL  :  : 
or  generally, 

solid  A G     :     solid  AL   :  : 


A 


\ 


15 


AE 


A  I. 


THEOREM     XII. 

Two  regular  parallelopipedons,  having  the  same  altitude,  are  to 
each  other  as  their  bases. 

Let  the  parallelopipe- 
dons AG  J  AK,  have  the 
same  altitude  AE ;  then 
will  they  be  to  each 
other  as  their  bases  AC, 
AN. 

Having  placed  the  two 
solids  by  the  side  of  each 
other,  as  the  figure  re- 
presents, produce  the 
plane  ONKL  until  it 
meets  the  plane  DCGH 
in   PQ;  you   will    thus 


144 


G  E  0  I\l  E  T  R  Y . 


Of    Par  all  p1  0  pi  pedona 


\^ 


M- 


H 


N 


have    a   third   parallelo- 

pipedon  A  Q,  which  may 

be  compared  with  each 

of  the  parallelopipedon? 

AG.AK.     The  two  sol- 
ids AG,  AQ,  having  the 

same  base  AEHD,  are 

to  cacli  otlier   as    their 

altitudes    AB,    AO ;    in 

like    manner,    the    two 

solids  AQ  AK,  having 

the  same   base  AOLE, 

are  to  each  other  as  their 

altitudes  AD,  AM. 

Hence,  we  have  the  two  proportions, 

solid  AG     :     solid  AQ     :  :     AB 
solid  AQ     :     solid  AK     :  :     AD 


\ 


W 


AO, 
AM. 


Muhiplying  together  the  corresponding  terms  of  these  pro- 
portions, and  omitting  the  common  multiplier  solid  AQ,  we  have 

solid  AG     :     solid  A K     ::     ABxAD     :     AOxAM. 
But  ABx  AD  represents  the   base  ABCD;  and  AOxAM 
represents  the  base  AMNO:  hence,  two  rectangidar  parallel- 
opipedons  of  the  same  altitude  are  to  each  other  as  their  bases. 

THEOREM      XIII. 

Any  two  rectangular  parallelopidedons  are  to  each  other  as  th* 
products  of  their  three  dimensions. 
For,  having  placed  the  two  solids  AG,  AZ,  (see  next  figure) 
so  that  their  surfaces  have  the  common  angle  BAE,  produce 
the  planes  necessary  for  completing  the  third  parallelopipedon 
AK,  having  the  same  altitude  vrith  the  parallelopipedon  AG 
By  the  last  proposition  we  shall  have  the  proportion. 


B  0  O  h.    V  i  . 


146 


Of    Parallelopipcdons. 


solid  AG     :     solid  AK    :  :     ABCD 


E 


AMNO 


But  the  two  paral- 
lelopipcdons AK,  AZ, 
having  the  same  base 
AMNO,  are  to  each 
other  as  their  altitudes 
AE,  AX ;  hence,  we 
have 


t 


N 


i 


tG 


solid  AK     :     solid  AZ     :  :     AE     :     AX. 

Multi')lying  together  the  corresponding  terms  of  these  pro- 
[wrtions,  and  omitting  in  the  result  the  common  mulliplioi 
solid  AK,  we  shall  have 

solid  AG  .  solid  AZ  :  :  ABCDxAE  :  AMNOxAX. 

Instead  of  the  bases  ABCD  and  AMNO,  put  ABxAD 
and  AOxAM,  and  we  have 

solid  AG  :  solid  AZ  •:  ABxADxAE  :  AOxAMxAX. 

Hence,  any  two  rectangidar  parallelopipcdons  are  to  each 
other  as  the  product  of  their  three  dimensions. 

Sch.  We  are  consequently  authorized  to  assume,  as  the 
measure  of  a  rectangular  parallelopipedon,  the  product  of  ito 
three  dimensions. 

In  order  to  comprehend  the  nature  of  this  measurement,  ii 

is  necessary  to  reflect,  that  the  number  of  linear  units  in  one 
13 


14G 


GEOMETRY. 


Of    Parallelopipedons, 


dimention  of  the  base  multiplied  by  the  number  of  linear  units 
of  the  other  dimension  of  the  base,  will  give  the  number  of 
superficial  units  in  the  base  of  the  parallelopipedon  (Bk.  1 V 
Th.  vi.  Sch).  For  each  unit  in  height,  there  are  evidently  a^ 
many  solid  units  as  there  are  superficial  units  in  the  base. 
Therefore,  the  product  of  the  number  of  superficial  units  in  the 
base  multiplied  by  the  number  of  linear  units  in  the  altitude 
is  the  number  of  solid  units  in  the  parallelopipedon. 

If  the  three  dimensions  of  another  parallelopipedon  are  valuod 
according  to  the  same  linear  unit,  and  multiplied  together  ip 
the  same  manner,  the  two  products  will  be  to  eacti  other  an. 
the  solids,  and  will  serve  to  express  their  relative  magnitude 

Let  us  illustrate  this  by  an  example. 

Let  ABCD  be  the  base  of  a 
parallelopipedon,  and  suppose 
AB  =  4  feet,  and  BC=3  feet. 
Then  the  number  of  square  feet 
m  the  base  ABCD  will  be  equal 
to  3x4=12  square  feet 

Therefore,  12  equal  cubes  of  1 
fact  each,  may  be  placed  by  the 
side  of  each  other  on  the  base.  If  the  parallelopipedon  be  J 
foot  in  height,  it  will  contain  12  cubic  feet ;  were  it  2  feet  in 
height  it  would  contain  two  tiers  of  cubes,  or  24  cubic  feet; 
were  it  3  feet  m  height,  it  would  contain  three  tiers  of  tubes, 
or  36  cubic  feet. 

The  magnitude  of  a  solid,  its  volume  or  extent,  forms  what 
is  called  its  solidity ;  and  this  word  is  exclusively  employed 
to  det'ignate  the  measure  of  a  solid  ;  thus,  we  say  the  s  )lidity 
of  a  rectangular  parallelopipedon  is  equal  to  the  product  of  its 
base  by  its  altitude,  or  to  the  prodrjct  of  its  three  dimensions 


BOOK      VI. 


147 


Of    Paralie  1  opipedons. 


As  ihc  cube  has  all  its  three  dimensions  equal,  if  the  side 
is  1,  the  solidity  will  be  1  x  1  X  1  =  1  ;  if  the  side  is  2,  the 
solidity  will  be  2x2x2  —  8;  if  the  side  is  3,  the  solidity 
will  be  3x3>:3=27;  and  so  on:  hence,  if  the  sides  of  a 
i^crios  of  cubes  are  to  each  other  as  the  numbers  1,  2,  3.  Sic, 
the  Qubes  themselves,  or  their  solidities,  will  l>e  as  the  num- 
bers 1,  8,  27,  Slc.  Hence  it  is,  that  in  arithmetic,  the  cube  of 
a  number  is  the  name  given  to  a  product  which  results  from 
three  factors,  each  equal  to  this  number. 


THEOREM     XIV. 

If  a  parallelopipedon,  a  prism,  and  a  cylinder,  have  equivaUiii 

bases  and  equal  altitudes,  they  will  be  equivalent. 

Let  F—ABCD,  be  a  parallelopipedon;  F— ABODE,  a 
prism ;  and  D — ABC,  a  cylinder,  having  equivalent  bases 
and  equal  altitudes  :  then  will  they  be  equivalent. 


'^ 


£ 


\ 


IT 


^ 


B   C 


r 


For,  since  their  bases  are  equivalent  they  will  contain  the 
same  number  of  units  of  surface  (Bk.  IV.  Def.  9).  Now 
for  e.ich  unit  of  height  there  will  be  one  tier  of  equal  cubei 
in  each  solid,  and  since  the  altitudes  are  equal,  the  number  o 
tiers  ill  each  solid  will  be  equal :  hence,  the  solidities  will  bt 
equal,  and  therefore  the  solids  will  be  equivalent. 

Cot    Hence,  we  conclude,  that  the  solidity  of  a  prism  < 
cylinder  is  equal   t  >  the  area  of  its  base  mtdtiplied   by  i\ ; 
altitude. 


us 


GEOMETRY. 


()f    Triangular    I'yramids. 


THEOREM      XV. 

Two  triangular  pyramids,  hiving  equivalent   bases  and  equal 

altitudes,  are  equivalent,  or  equal  in  solidity. 


Let  their  equivaleul  bases,  ABC,  abc,  be  situated  in  the 
same  plane,  and  let  AT  be  their  common  altitude.  If  they 
arc  not  equivalent,  let  S — abc  he  the  smaller;  and  suppose 
Aa  to  be  the  altitude  of  a  prism,  which,  having  ABC  for  its 
base,  is  equal  to  their  difference. 

Divide  the  altitude  AT  into  equal  parts  Ax,  cry,  yz,  &c. 
each  less  than  Aa,  and  let  k  be  one  of  those  parts  :  through 
the  points  of  division  pass  planes  parallel  to  the  plane  of  the 
bases :  the  corresponding  sections  formed  by  these  planes  in 
the  two  pyramids  wilj  be  respectively  equivalent,  namely 
DEF  to  def,  GHI  to  gki,  &c.  (Th.  v.  Cor.) 


BOOK     VI.  149 


Of    T  r  i  angular    Pyramida. 


This  being  granted,  upon  the  triangles  ABCj  DEF,  GUI, 
(fee,  taken  as  bases,  construct  extejrior  prisms  hanng  ioj 
edges  the  parts  AD,  DG,  GK,  &c.,  of  the  edge  SA  ;  in  like 
manner,  on  bases  dcf,  ghi,  klm,  &c ,  in  the  second  pyramid 
construct  interior  prisms,  having  for  edges  the  corresponding 
parts  of  Sa.  It  is  plain  that  the  sum  of  the  exterior  prisms  o\ 
the  pyramid  5 — ABC  will  be  greater  than  the  pyramid;  while 
the  sum  of  the  interior  prisms  of  the  pyramid  <S — abc,  will  be 
less  than  the  pyramid.  Hence,  the  diflerence  between  these 
sums  will  be  greater  than  the  difference  between  the  pyramids. 

Now,  beginning  with  the  bases  ABC,  abc,  the  second  ex- 
terior prism  JSFB — G  is  equivalent  to  the  first  interior  prism 
efd — a,  because  they  have  the  same  altitude  k,  and  their  bases 
DEF,  def,  are  equivalent;  for  like  reasons,  the  third  exterior 
prism  JUG — K,  and  the  second  interior  prism  hig — d,  are 
equivalent ;  the  fourth  exterior  and  the  third  interior ;  and  so 
on,  to  the  last  of  each  scries.  Hence,  all  the  exterior  prisms 
of  the  pyramid  5 — ABC,  excepting  the  first  prism  BCA — D, 
have  equivalent  corresponding  ones  in  the  interior  prisms  oi 
the  pyramid  5" — abc:  hence,  the  prism  BCA — D  is  the  differ- 
ence between  the  sum  of  all  the  exterior  prisms  of  the  pyramid 
S — ABC,  and  of  the  interior  prisms  of  the  pyramid  5 — abc 
But  this  difl'crcucc  has  already  been  proved  to  be  greater  thar. 
that  of  the  two  pyramids:  which,  by  supposition,  difl!*er  by 
the  prism  a — ABC:  hence,  the  prism  BCA — Z),  must  be 
erreater  than  the  prism  a — ABC.  But  in  reality  it  is  less,  foi 
they  have  the  same  base  ABC,  and  the  altitude  Ax,  of  the 
first,  is  less  than  Aa,  the  altitude  of  the  second.  Hence,  the 
supposed  inequality  between  the  two  pyramids  cannot  exist; 
lience,  the  two  p^Tamids;  S — ABC,  S — abc,  having  equal  al 
titudes  and  equivalent  bases,  are  themselves  equivalent. 
13* 


150 


GEOMETRY. 


Of    Triangular    Pyramids 


THEOREM     XVI. 

Every  triangular  pyromid  is  a  third  part   of  a  triangulai  pris 
which  has  an  equal  base  and  the  same  altitude. 

Lei  F — ABC  be  a  trian- 
gular pyramid,  ^Z]C — DEF 
a  triangular  prism  of  the 
same  base  and  the  same  al- 
titude :  the  pyramid  will  bo 
equal  to  a  third  of  the  prism. 

Cut  off  the  pyramid  F — 
ABC  from  the  prism,  by  the 
plane  FAC ;  there  will  re- 
main the  solid  F — A  CDE, 
which  may  be  considered 
as  a  quadrangular  pyramid,  whose  vertex  is  F,  and  wnosc 
base  is  the  parallelogram  ACDE,  Draw  the  diagonal  CE, 
and  pass  the  plane  FCE,  which  will  cut  the  quadrangular 
pyramid  into  two  triangular  ones,  F-ACE,  F-CDE.  These 
two  triangular  p}Tamids  have  for  their  common  altitude  the 
perpendicular  let  fall  from  F  on  the  plane  ACDE;  and 
their  bases  are  also  equal,  being  halves  of  the  parallelogram 
AD:  hence,  the  pyramid  F-ACE,  and  the  pjTamid  F-CDE, 
are  equivalent  (Th.  xv). 

But  the  pyTamid  F — CDE,  and  the  pyramid  F — ABC,  have 
equal  bases,  ABC,  DEF ;  they  have  also  the  same  altitude, 
namely,  the  distance  between  the  parallel  planes  ABC,  DEF, 
hence,  the  two  pyramids  are  equivjilent.  Now,  the  pyramid 
F — CDE  has  already  been  proved  equivalent  to  F — ACE; 
hence,  the  three  pyramids  F-^ABC,  F—CDE,  F—ACE, 
which  compose  the  prism  ABC — DEF  are  all  oquivaleni 


BOOK     VI. 


15J 


Soli«lity    of    the    Py r amid 


Hence,  the  pjTamid  F—ABC  is  the  third  part  of  the  prism 
ABC — DEF,  which  has  an  equal  base  and  the  same  altitude. 
Cor.  The  solidity  of  a  triangular  pyramid  is  equal  to  a  third 
part  of  the  product  of  its  base  by  its  altitude. 

THEOREM    XVII. 

The  solidity  of  every  pyramid  is  equal  tc  the  hase  multiplied  by 
a  third  of  the  altitude. 

Let  <S — ABCDE  be  a  pyramid. 

Pass  the  planes  SEBy  SEC  through 
the  diagonals  EB,  EC ;  the  polygonal 
p}Tamid  5 — ABCDE  will  be  divided 
into  several  triangular  pyramids  all 
having  the  same  altitude  SO.  But 
each  of  these  pyramids  is  measured  by 
midtiplying  its  base  ABE,  BCE,  or 
CDE,  by  the  third  part  of  its  altitude 
SO  (Th.  xvi.  Cor);  hence  the  sjm 
of  those  triangular  pyramids,  or  the  polygonal  pyramid 
iS — ABCDE,  will  1)0  measured  by  the  sum  of  the  triangles 
ABE,  BCE,  CDE,  or  the  polygon  ABCDE,  multiplied  by 
one  third  of  SO. 

Cor.   1.  Every  pjTamid  is  the  third  part  of  the  prism  \rhich 
has  the  same  base  and  tlic  same  altitude. 

Cor.  2.  Two  p}Tamids  having  the  same   altitude,  are  to 
each  other  as  their  bases. 

Cor.  3.  Two  pjTamids  having  equivalent  bases,  are  to  eauh 
other  as  their  altitudes. 

Cor.  4.  Pyramids  are  to  each  other  as  the  products  of  their 
bases  by  their  altitudea 


i52  GEOMETRY 


Solidity    of    the    Cone 


THEOREM     XVIII. 

The  solidity  of  a  cone  h  equal  to  one  third  of  the  product  oftlie 
base  multiplied  by  the  altitude. 

Let  ABODE  be  the  base,  5"  the 
vertex,  and  SO  the  altitude  of  the 
cone :  then  will  its  solidity  be  equal 
to  one  third  the  product  of  its  base 
by  its  altitude  SO. 

Inscribe  in  the  base  of  the  cone 
any  regular  polygon,  ABODE,  and 
join  the  vertices  A,  B,  0,  &c.,  with 
the  vertex  S,  of  the  cone  ;  then  will 
there  be  inscribed  in  the  cone  a  right  pyramid,  having 
for  its  base  the  polygon  ABODE.  The  solidity  of  this 
pyramid  is  equal  to  one  third  of  the  base  multiplied  by  the 
altitude  (Th.  xvii). 

Let  now,  the  number  of  sides  of  the  polygon  be  indefinitely 
increased :  the  polygon  will  then  become  equal  to  the  circle, 
and  the  pyramid  and  cone  will  coincide  and  become  equal. 
But  the  solidity  of  the  pyramid  will  still  be  equal  to  one  third 
of  the  product  of  the  base  multiplied  by  the  altitude,  whatever 
be  the  number  of  sides  of  the  polygon  which  forms  its  base ; 
hence,  the  solidity  of  the  cone  is  equal  to  one  third  of  the 
product  of  its  base  multiplied  by  its  altitude. 

Cor.  I.  A  cone  is  the  third  part  of  a  cylinder  havmg  the 
same  base  and  the  same  altitude ;  whence  it  follows : 

Ist,  That  cones  of  equal  altitudes  are  to  each  other  as  thei: 
based. 

2nd,  That  cones  of  equal  bases  are  to  each  other  ae  their 
altitudes. 


BOOK      VI 


153 


Of    PrisniP, 


Cor.  2.  The  solidity  of  a  cone  is  equivalent  to  the  solidity 
of  a  pjTaniid  having  an  equivalent  base  and  the  same  altitude. 


THEOREM    XIX 

Similar  pnsms  arc  to  each  other  as  the  cubes  of  their  homologous 
edges. 


Let  ABC—D,  EFG—H  be 
flimilar  prisms :  then  we  shall 
have 


<& 


B_C 
solid  AD     :     solid  EH     :  :     AB^     : 

or  solid  AD     :     solid  EH     ::     CD^     :    HG^; 

or,  the  solids  will  be  to  each  other  as  the  cubes  of  any  othci 
of  their  homologous  edges. 

For,  the  solids  are  to  each  other  as  the  products  of  theii 
bases  and  altitudes  (Th.  xiv.  Cor.),  that  is, 

solid  ABC-D  :  solid  EFG-H  :  :  ABCx  CD  :  EFGxGH. 

Ijut  the  bases  being  similar  polygons  are  to  each  other  as  the 
squares  of  their  like  sides  (Bk.  IV.  Th.  xxi) ;  that  is, 

ABC     :     EFG     :  :     AS'     :     Ep, 
therefore, 
solid  ABO-D  :  solid  EFG-H  :  •  Iff  x  CD  :  EF^xGH 


154 


GEOIMETRY 


Of    Prisms 


But  since  the  solids  are  simi- 
lar, the  parallelograms  BD  and 
FII  nrv  similar  (Def.  3) :  hence, 
CD  and  GH  are  proportional  to 
BC  and  FG,  and  consequently 
to  AB  a.nd  EF:  hence,  we  have, 


B~C 


AB'^xAB  :  EF'^xEF, 


solid  ABC-D  :  solid  EFG-H 
that  is, 

solid  ABC-D     :     solid  EFG-H    :  :     aT3^     :      W^; 

and  in  a  similar  manner  it  may  be  shown  that  the    solida 
are  to  each  other  as  the  cubes  of  any  other  homologous  edges. 

Cor,  Since  cylinders  arc  to  each  other  as  the  product  of 
their  bases  and  altitudes  (Th.  xiv.  Cor.),  it  follows  that  similar 
cylinders  are  to  each  other  as  the  cubes  of  the  linear  dimen 
sions. 

THEOREM    XX. 

Every  section  oj  a  sphere,  made  by  a  plane,  is  a  circle. 

Let  AMB  be  a  section,  made  by 
a  plane,  in  the  sphere  whose  cen- 
tre is  C. 

From  the  centre  C  draw  CO, 
perpendicular  to  the  plane  AMB, 
and  also  draw  the  lines  CA,  CM, 
&c.,  to  the  points  of  the  curve 
AMB,  which  terminate  the  sec- 
tion, and  join  OA,  OM,  &c. 


BOOK      VI. 


165 


Of    the    Sphere 


Then,  since  CO  is  pcrdcndic- 
ular  to  ilic  piano  AMB,  the  an- 
gles COA,  COM  &c.,  will  be 
right  angles,  and  since  the  radii 
of  the  sphere  arc  all  equal,  the 
right  angled  triangles  CA  O,  COM, 
&c.,  will  have  the  hypothenuses 
equal,  and  the  side  CO  common  : 
hence,  the  remaining  sides  will  be  equal  (Bk.  I.  Th.  xix). 
Therefore,  all  lines  drawn  from  O  to  any  point  of  the  curvo 
AMB  are  equal :  hence  AMB  is  a  circle. 

Cor.  1.  If  the  section  passes  through  the  centre  of  the 
sphere,  its  radius  will  be  the  radius  of  the  sphere  :  hence,  all 
great  circles  are  equal. 

Cor.  2.  Two  great  circles  always  bisect  each  other;  for 
their  common  intersection,  passing  through  the  centre,  is  a 
diameter. 

Cor.  3.  Every  great  circle  divides  the  sphere  and  its  sur- 
face into  two  equal  parts  :  for,  if  the  two  hemispheres  were 
separated  id  afterwards  placed  on  the  common  base,  with 
their  convexities  turned  the  same  way,  the  two  surfaces  would 
exactly  coincide,  no  point  of  the  one  being  nearer  the  centi«> 
tlian  any  point  of  the  other. 

Cor.  4.  The  centre  of  a  small  circle,  and  that  of  the  sphere, 
are  in  the  same  straight  line,  perpendicular  to  the  plane  of  the 
small  circle 


Cor   5.  Small  circles  are  the  less  the  fartlior  they  lie  from 


56 


G  E  O  M  E  T  R  y 


Of    the    Sphere 


the  centre  of  the  sphere  ;  for  the  greater  CO  is,  the  less  is 
the  chord  AB,  the  diameter  of  the  small  circle  AMB. 

THEOREM      XXI. 

Everp  plane  perpendicular  to  a  radius  rt  its  extremity  is  law 
gent  to  the  sphere. 

Let  FAG  be  a  plane  perpen- 
dicular to  the  radius  OA^  at  its 
extremity  A.  Any  point  M,  in 
this  plane,  being  assumed,  and 
OM,  AM,  being  drawn,  the  an- 
gle 0AM  will  be  a  right  angle, 
and  hence,  the  distance  OM  will 
be  greater  than  OA.  Hence, 
the  point  M  lies  without  the  sphere  ;  and  as  the  same  can  be 
showTi  for  every  other  point  of  the  plane  FA  G,  this  plane  can 
have  no  point  but  A  common  to  it  and  the  surface  of  the 
sphere  ;  hence  it  is  a  tangent  plane  (Def.  31). 

Sch.  In  the  same  way  it  may  be  shown,  that  two  spheres 

have  but  one  point  in  common,  and  therefore   touch  each 

other,  when  the  distance  between  their  centres  is  equal  to  the 

sum,  or  the   difference   of  their  radii ;  in  either   case,  the 

ontrcs  and  the  point  of  contact  lie  in  the  same  straight  line. 


THEOREM     XXII. 

If  a  regular  semi-polygon  he  revolved  about  a  line  passing 
through  the  centre  and  the  vertices  of  two  opposite  angles^  the 
surface  described  by  its  perimeter  will  be  equal  to  the  axis  multi 
vlied  by  the  circumference  of  the  inscribed  circle. 


BOOK     VI 


157 


Of    t  h  e    Sphere 


Suppose  the  regular  semi-polygon 
ABODE  to  be  revolved  about  the  line 
AF  as  an  axis:  then  will  the  surface 
described  by  its  perimeter  be  equal  to 
AF  multiplied  by  the  circumference  of 
the  inscribed  circle. 

From  E  and  Z),  the  extremities  of 
one  of  the  equal  sides,  let  fall  the  per- 
pendiculars EH,  DI,  on  the  axis  AFy 
and  from  the  centre  O,  draw  ON  per- 
pendicular to  the  side  DE:  ON  will  then  be  the  radius  of  the 
inscribed  circle  (Bk.  IV.  Prob.  x). 

Ijct  us  first  find  the  measure  of  the  surface  described  by 
one  of  the  equal  sides,  as  DE. 

From  N,  the  middle  point  of  DE,  draw  NM  perpendicular 
to  the  axis  AF,  and  tlirough  E,  draw  EK,  parallel  to  it,  meet- 
ing MN  in  S. 

Then,  since  EN  is  half  of  ED,  NS  will  be  half  of  DK 
(Bk.  IV.  Th.  xiii) :  and  hence,  NM  is  equal  to  half  the  sum 
oi  EH+DL 

But,  since  the  circumferences  of  circles  are  to  each  other  as 
their  diameters  (Bk.  IV.  Th.  xxiv),  or  as  their  radii,  the 
halves  of  the  diameters,  we  shall  have  the  circumference  do- 
scribed  by  the  point  N,  equal  to  half  the  sum  of  the  circum- 
ferences described  by  the  points  D  and  E. 

But  in  the  revolution  of  the  polygon  the  line  ED  describes 
the  siurface  of  the  frustum  of  a  cone,  the  measure  of  which  is 
equal  to  DE  multiplied  into  half  the  sum  of  the  circumfe- 
rences of  the  two  bases  (Th.  ix) ;  that  is,  equal  to  DE  into 

the  circumference  described  by  the  point  N 
14 


158 


GEOMETRY 


Of    the    Sphere 


But,  the  triangle  ENS  is  similar  to 
SNT  (Bk.  IV.  Th.  xviii),  and  also  to 
EDKy  and  since  TNS  is  similar  to 
ONM,  it  follows  that  EDK  and  ONM 
aro  similar ;  hence, 


ED     :     EK  or  ///     :  :     ON    :     NM, 
or    ED  :  HI  :  :  circumference  ON  :  circumference  MN. 
consequently, 

E D  X  circumference  MN=  HI  X  circumference  ON, 

that  is,  ED  multiplied  into  the  circumference  of  the  circle  de- 
scribed with  the  radius  NM,  is  equal  to  HI  into  the  circum* 
fcrence  of  the  circle  described  with  the  radius  ON^.  But  the 
former  is  equal  to  the  surface  described  by  the  line  ED  in  the 
revolution  of  the  polygon  about  the  axis  AF;  hence,  the  latter 
is  equal  to  the  same  area ;  and  since  the  same  may  be  shown 
for  each  of  the  other  sides,  it  is  plain  that  the  surface  des- 
cribed by  the  entire  perimeter  is  equal  to 
(FH-\-HI+IP-\-PQ+QA)xcirf.  ON=AFxcirf.  ON. 

Cor.  The  surface  described  by  any  portion  of  the  perim* 
eter,  as  EDO,  is  equal  to  the  distance  between  the  two  per- 
pendiculars let  fall  from  its  extremities,  on  the  axis,  multiplisd 
by  the  circumference  of  the  inscribed  circle.  For,  the  sur- 
face described  by  DE  is  equal  to  ///x  circumference  ON. 
and  the  surface  described  by  DC  is  equal  to  /Pxcircumfe- 


BOOK     VI  . 


159 


Of    the    Sphere. 


ronce  ON:  hence,  the  surface  described  by  JE^D+DC,  is  equal 
(o  (///H-/P)x  circumference  ON,  ot  equal  to  HP  xciicum' 
ference  ON. 


THEOREM     XXIII. 

The  surface  of  a  sphere  is  equal  to  the  product  of  its  diameter 
hy  the  circumference  of  a  great  circle. 

JiCt  ABODE  be  a  semicircle.  In- 
jcribe  in  it  any  regular  semi-polygon, 
and  from  the  centre  0  draw  OF  per- 
pendicular to  one  of  the  sides. 

Let  the  semicircle  and  the  semi- 
polygon  be  revolved  about  the  axis 
AE:  the  semicircumference  ABODE 
will  describe  the  surface  of  a  sphere 
(Def.  26) ;  and  the  perimeter  of  the 
semi-polygon  will  describe  a  surface 
which  has  for  its  measure  -A^x  cir- 
cumference OF  (Th.  xxii) ;  and  this  will  be  true  whatever  bo 
the  number  of  sides  of  the  polygon.  But  if  the  number  of 
sides  of  the  polygon  be  indefinitely  increased,  its  perimeter 
will  coincide  ^v^th  the  circumference  ABODE,  the  perpen- 
dicular OF  will  become  equal  to  OE,  and  the  surface  de- 
scribed by  the  perimeter  of  the  semi-polygon  will  then  be  the 
same  as  that  described  by  the  semicircumference  ABODE 
Hence,  the  surface  of  the  sphere  is  equal  to  ^J^^xcircum 
ference  OE. 

Cor  Since  the  area  of  a  great  circle  is  equal  to  the  product 
of  its  circumference  by  half  the  radius,  or  by  one-fourth  of 
the  diameter  (Bk.  IV.  Th.  xxvii),  it  follows  that  the  surface 
of  a  sphere  is  equal  to  four  of  its  great  circl'DS. 


160 


GEOMETRY. 


Of    the     Zone 


THEOREM     XXIV. 

The  surface  of  a  zone  is  equal  to  its  altitude  multiplied  by 
the  circumference  of  a  great  circle. 

For,  tho  surface  described  by  any 
|X)rtion  of  the  perimeter  of  the  in- 
scribed polygon,  as  BC+CD  is  equal 
to  £//x  circumference  OF  (Th.  xxii. 
Cor).  But  when  the  number  of  sides 
of  the  polygon  is  indefinitely  increased, 
BC-\'CD,  becomes  the  arc  BCD,  OF 
becomes  equal  to  OA,  and  the  surface 
described  by  BC-\CD,  becomes  the 
surface  of  the  zone  described  by  the 
arc  BCD:  hence,  the  surface  of  the 
zone  is  equal  to  £i/xcircmnference 
OA. 

Sch.  1.  When  the  zone  has  but  one  base,  as  the  zone  do 
scribed  by  the  arc  ABCD,  its  surface  will  still  be  equal  to 
the  altitude  AE  multiplied  by  the  circumference  of  a  great 
circle. 

Sch.  2.  Two  zones  taken  in  the  same  sphere,  or  in  equal 
spheres,  are  to  each  other  as  their  altitudes ;  and  any  zone  is 
to  the  surface  of  the  sphere  as  the  altitude  of  the  zone  is  to 
tho  diameter  of  the  si  here. 


THEOREM      XXV. 

The  solidity  of  a  sphere  is  equal  to  one  third  of  the  product  yf 
the  surface  multiplied  hy  the  radius. 
For,  conceive  a  polyedron  to  bo   inscribed   in  the   ?phore. 


BOOK      V  i  . 


16] 


Of    the    Sphere 


This  polyedron  may  be  considered  as  formed  of  pyramids,  each 
having  for  its  vertex  the  centre  of  the  sphere,  and  for  its  base 
one  of  the  faces  of  the  polyedron.  Now,  the  solidity  of  each 
pyramid,  will  be  equal  to  one  tliird  of  the  product  of  its  base 
by  its  altitude  (Th.  xvii). 

But  if  we  suppose  the  faces  of  the  polyedron  to  be  conlinu- 
ally  diminished,  and  consequently,  the  number  of  the  pyra- 
mids to  be  constantly  increased,  the  polyedron  will  fmaily 
become  the  sphere,  and  the  bases  of  all  the  pyramids  will 
become  the  surface  of  the  sphere.  When  this  takes  place, 
the  solidities  of  the  pyramids  will  still  be  equal  to  one  third 
th3  product  of  the  bases  by  the  common  altitude,  which  y,  ill 
then  be  equal  to  the  radius  of  the  sphere. 

Hence,  the  solidity  of  a  sphere  is  equal  to  one  third  of  the 
product  of  the  surface  bv  the  radius. 


THEOREM     XXVI. 

The  surface  of  a  sphere  is  equal  to  the  convex  surface  of  the 
circumscribing  cylinder ;  and  the  solidity  of  the  sphere  is  two 
thirds  the  solidity  of  the  circumscribing  cylinder. 

Let  MPNQ  be  a  great  circle  of 
the  sphere ;  ABCD  the  circum- 
scribing square  :  if  the  semicircle 
PMQ,  and  the  half  square  PADQ, 
are  at  the  same  time  made  to  re- 
volve about  the  diameter  PQ,  the 
Beniicirclc  will  describe  the  sphere, 
vhilc  the  half  square  will  describe 
the  cylinder  circumscribed  about 
that  sphere. 

The  altitude  AD,  of  the  cylinder,  is  equal  to  the  diameter 
14* 


162 


G  K  0  ]\I  E  T  R  Y 


Of    the    Sphere. 


PQ;  llic  base  of  the  cylinder  is 
equal  to  llie  great  circle,  since  its 
diameter  ^i3  is  equal  MN;  hence, 
the  convex  surface  of  the  cylin- 
der is  equal  to  the  circumference 
of  the  great  circle  multiplied  by 
its  diameter  (Th.  ii).  This  meas- 
ure is  the  same  as  that  of  the  sur- 
face of  the  sphere  (Th.  xxiii) : 
hence  ilie  surface  of  the  sphere  is  equal  to  the  convex  sur- 
face of  the  circumscribing  cylinder. 

In  the  next  place,  since  the  base  of  the  circumscribing 
cylinder  is  equal  to  a  great  circle,  and  its  altitude  to  the  di- 
ameter, the  solidity  of  the  cylinder  will  be  equal  to  a  great 
circle  multiplied  by  a  diameter  ('J1i.  xiv.  Cor).  But  the  so- 
lidity of  the  sphere  is  equal  to  its  surface  multiplied  by  a  third 
of  its  radius  ;  and  since  the  surface  is  equal  to  four  great 
circles  (Th.  xxiii.  Cor.),  the  solidity  is  equal  to  four  great  cir- 
cles multiplied  by  a  third  of  the  radius  ;  in  other  words,  to 
one  great  circle  multiplied  by  four-thirds  of  the  radius,  oi 
by  two-thirds  of  the  diameter ,  hence,  the  sphere  is  two-thirds 
of  the  circimiscribirg  cyhndcr. 


BOOK     V 


163 


Appendix. 


APPENDIX 
OP      THE      FIVE      REGULAR      POLYEDRONS- 

A  regular  polyedron,  is  one  wliose  faces  are  all  equal  poly- 
gons, and  whose  polycdral  angles  are  equal.  There  are  fi^e 
Ruch  solids. 

1.  The  Tetraedron,  or  equilateral  p}Tamid,  is  a  solid  bounded 
t>y  four  equal  triangles. 


2.  The  kexacdron  or  cube,  is  a  solid,  boiir  dnd  by  six  equal 
squaros. 


r 


3    The  oclaedron,  is  a  solid,  bounded  by  eight  equal  o<xtn 
lateral  triangles. 


164 


GEOMETRY. 


Append 


4.  The  dodecacdron,  is  a  solid  bounded  by  twelve  oquaJ 
pentagons 


^^^ 


"^ 


5.  The  tcosacdron,  is    a   solid,   bounded   by  twenty  equa' 
equilateral  triangles. 


6.  The  regular  solids  may  easily  be  made  of  pasteboard. 

Draw  the  figures  of  the  regular  solids  accurately  on  paste 
board,  and  then  cut  through  the  bounding  lines  :  this  will  give 
fifmres  of  pasteboard  shnilar  to  the  diagrams.  Then,  cut 
the  other  lines  half  through  the  pasteboard,  after  which,  turn 
up  the  parts,  and  glue  them  together,  and  you  will  form  the 
bodies  which  have  been  described. 


ELEMENTS  OF  TRIGONOMETRY, 


INTRODUCTION 
SECTION  I. 

or    LOOAaiTHKS. 


1.  The  logarithm  of  a  number  is  the  exponent  of  the  power 
to  which  it  is  necessary  to  raise  a  fixed  number^  in  order  to 
produce  the.  first  number. 

This  fixed  number  is  called  the  base  of  the  system,  and  may 
be  any  number  except  1  :  in  the  common  system  10  is  assumed 
as  the  base. 

2.  If  we  form  those  powers  of  10,  which  are  denoted  by  entire 
exponents,  we  shall  have 

10°  =  1  10*  =  10       ,       lO'  =  1000 

10^  =  100     ,       10*  =  10000,  <fec.  (fee. 
From  the  above  table,  it  is  plain,  that  0,  1,  2,  3,  4,  (fee,  are  re- 
t^pcctively  the  logarithms  of  1,  10,  100,  1000,  10000,  &c.;  we 
also  see  that  the  loj^arithm  of  any  numbe.*  between  1  and   10  ia 
greater  than  0  -nd  less  than  1  :  thus 
Log  2=: 0.801030 


1(10  TRIGONOMETRY. 

Of   Logarithms. 

The  logarithm  of  any  number  greater  than  10,  and  less  than 
100,  is  greater  than  1  and  less  than  2  :  thus 
Log  60  =  1.698970 

The  logarithm  of  any  number  greater  than  100,  and  less  Ihau 
1000,  is  greater  than  2  and  less  than  3  :  thus 
Log  126  -=  2.100371,  <fec. 

If  the  above  principles  be  extended  to  other  numbers,  it  will 
appear,  that  the  logarithm  of  any  number,  not  an  exact  power 
of  ten,  is  made  up  of  two  parts,  an  entire  and  a  decimal  2mrt. 
The  entire  part  is  called  the  characteristic  of  the  logarithm^ 
and  is  always  one  less  than  the  number  of  places  of  figures  in  the 
given  number. 

3.  The  principal  use  of  logarithms,  is  to  abridge  numerical 
cx)mputations. 

Let  M  denote  any  number,  and  let  its  logarithm  be  denoted 
by  m  ;  also  let  N  denote  a  second  number  whose  logarithm  ia 
n  ;  then  from  the  definition  we  shall  have 

10"=  M  (1)  10°  =  N  (2) 

Multiplying  equations  (1)  and  (2),  member  by  member,  we 
have 

lO""''  =  MxJ^    or,  m+n  =  log  MX-^-'     hence. 

The  sum  of  the  logarithms  of  any  two  numbers  is  equal  to 
the  logarithm  of  their  2^roduct. 

Dividing  equation  (1)  by  equation   (2),  member  by  member, 

we  have 

^^m-n        M  .        M       . 

10       =  — -    or,    m — n  =  log  -^'.    hence. 

The  logarithm  of  the  quotient  of  two  numbers^  is  equal  to  the 
logarithm  of  the  dividend  diminished  by  the  logarithm  of  the 
divisor. 


INTRODUCTION.  107 


Of    LogarithmB, 


4.  Since  the  logarithm  of  10  is  1,  the  logarithm  of  the  product 
of  any  number  by  10,  will  be  greater  by  1  than  the  logarithm  oj 
that  number  ;  also,  the  logarithm  of  any  number  divided  by  10^ 
xcill  be  less  by  1  than  the  logarithm  of  that  number. 

Similarly,  it  may  be  shown  that  the  logarithm  of  any  number 
multiplied  by  a  hundred,  is  greater  by  2  than  the  logarithm  of 
that  number,  and  the  logarithm  of  any  number  divided  by  100 
IB  less  by  2,  than  the  logarithm  of  that  number,  and  so  on. 


EXAMPLES. 

log  82Y 

is 

2.514548 

log  32.7 

a 

1.514548 

log  3.27 

u 

0.514548 

log  .327 

u 

1.514548 

log  .0327 

i( 

2.514548 

t\om  the  above  examples,  we  see,  that  in  a  number  composed 
of  an  entire  and  decimal  part,  we  may  change  the  place  of  thf 
decimal  point  without  changing  the  decimal  part  of  the  logarithm , 
but  the  characteristic  is  diminished  by  1  for  every  j^lf^ce  that  ths 
decirtuil  ])oint  is  removed  to  the  left. 

Tn  the  logarithm  of  a  decimal,  tlie  characteristic  becomes  nega- 
tive, and  is  numerically  1  greater  than  the  number  of  ciphers  im- 
mediately after  the  decimal  point.  The  negative  sign  extends 
only  to  the  characteristic,  and  is  written  over  it  as  in  the  exam- 
ples given  above. 

TABLE    OF    LOQARITUMS. 

5.  A  table  of  logarithms,  is  a  table  in  which  are  wntten  the 
logarithms  of  all  numbers  between  1  and  some  given  number. 
The  logarithms  of  all  numbers  between  1  and  10,000  are  giver 


168  TRIGONOMETRY. 

Of  Logarithms. 

in  the  annexed  table.  Since  rules  have  been  given  for  determin- 
ing the  characteristics  of  logarithms  by  simple  inspection,  it  has 
not  been  deemed  necessary  to  write  them  in  the  table,  the  deci- 
mal part  only  being  given.  The  characteristic,  however,  is  given 
for  all  numbers  less  than  100. 

The  left  hand  column  of  each  page  of  the  table,  is  the  column 
of  numbers,  and  is  designated  by  the  letter  N ;  the  logarithms 
of  these  numbers  are  placed  opposite  them  on  the  same  hori- 
zontal line.  .  The  last  column  on  each  page,  headed  D,  shows  the 
difference  between  the  logarithms  of  two  consecutive  numbers. 
This  difference  is  found  by  subtracting  the  logarithm  under  the 
column  headed  4,  from  the  one  in  the  column  headed  5  in  the 
9ame  horizontal  Hne,  and  is  nearly  a  mean  of  the  differencea 
of  any  two  consecutive  logarithms  on  the  line. 

6.   To  find  from  the  table  the  logarith0n  of  any  number. 

If  the  number  is  less  than  100,  look  on  the  first  page  of  the 
table,  in  the  column  of  numbers  under  N,  until  the  number  is 
found  :  the  number  opposite  is  the  logarithm  sought :  Thus 
log  9  =  0.954243 

V.    When  the  number  is  greater  than  100  and  less  than  10000. 

Find  in  the  column  of  numbers,  the  first  three  figures  of  the 
given  number.  Then  pass  across  the  page  along  a  horizontal 
line  until  you  come  into  the  column  under  the  fourth  figure  of 
the  given  number :  at  this  place,  there  are  four  figures  of  the 
required  logarithm,  to  which  two  figures  taken  from  the  column 
marked  0,  are  to  be  prefixed. 

If  the  four  figures  already  found  stand  opposite  a  row  of  six 
figures  'in  the  column  marked  0,  the  two  left  hand  figures  of 
the  six,  are  the  two   to  be  prefixed ;  but  if  they  stand  opposite 


I  N  T  R  0  D  U  C  1  I  O  N.  109 

Of   Logarithmfl. 

a  row  of  only  four  6gures,  you  ascend  the  column  till  you  find 
a  row  of  six  figures ;  the  two  left  hand  figures  of  this  row  are 
the  two  to  be  prefixed.  If  you  prefix  to  the  decimal  part  thus 
for.nd,  the  characteristic,  you  will  have  the  logarithm  sought : 
Thus, 

log     8979  =  3.953228 

log  .08979  =  2.953228 
If  however  in  passing  back  from  the  four  figures  found,  to  the 
0  column,  any  dots  be  met  with,  the  two  figures  to  be  prefixed 
must  be  taken  from  the  horizontal  line  directly  below  :    Thus, 

log    3098  =  3.491081 

log  30.98  =  1.491081 
If  the  logarithm  falls  at  a  place  where  the  dots  occur,  0  musl 
be  written  for  each   dot,  and  the  two  figures  to  be  prefixed  aro 
as  before   tiiken  from  the  line  below  :    Thus, 

log   2188  =  3.340047 

log  .2188  =  1.340047 

8.    When  the  number  exceeds  10,000. 

The  characteristic  is  determined  by  the  rules  already  given. 
To  find  the  decimal  part  of  the  logarithm.  Place  a  decimal 
point  after  the  fourth  figure  from  the  left  hand,  converting  the 
^ven  number  into  a  whole  number  and  decimal.  Find  the  loga- 
rithm of  the  entire  part  by  the  rule  just  given,  then  take  from 
the  right  hand  column  of  the  page,  under  D,  the  number  on  the 
same  horizontal  line  with  the  logarithm,  and  multiply  it  by  the 
d<!cimal  part ;  add  the  product  thus  obtained  to  the  logarithm  al- 
ready found,  and  the  sum  will   be  the  logarithm  sought. 

If,  in  multiplying  the  number  taken  from  the  column  D,  the 
decimal  part  of  the  product  exceeds  .5  let  1  be  added  to  the  ep- 
15 


170  TRIGONOMETRY 

Of    Logarithm  fl. 

tire  part;  if  it  is  less  than  .5  the  decimal  part  of  the  product  is 
neglected. 

EXAMPLE. 

To  find  log  672887. 

The  characteristic  is  5. ;  placing  a  decimal  point  after  the 
fourth  figure  from  the  left,  we  have  6728.87.  The  decimal  part 
of  the  log  6728  is  .827886  and  the  corresponding  number  in  the 
column  D  is  G5  ;  then  65X.87  =  56.55,  and  since  the  decimal 
part  exceeds  .5,  we  have  57  to  be  added  to  827886,  which  gives 
.827943 

or     log  672887      =  5.827943 
Similarly  log  .0072887  =  2.827943 

The  last  rule  has  been  deduced  under  the  supposition  that  the 
difference  of  the  numbers  is  proportional  to  the  difference  of 
their  logarithms,  which  is  suflBciently  exact  within  the  narrow 
limits  considered. 

In  the  alx)ve  example,  65  is  the  difference  between  the  loga- 
rithm of  672900  and  the  logarithm  of  672800,  that  is,  it  is  the 
difference  between  the  logarithms  of  two  numbers  which  differ  by 
100. 

We  have  then  the  proportion  100  :  87  :  65  .  56.55,  the 
number  to  be  added  to  the  logarithm  already  found. 

9.  To  find  from  the  table  the  number  corresp^yriding  to  a 
ghen  logarithm. 

Search  in  the  columns  of  logarithms  for  the  decimal  part  of 
the  given  logarithm  :  if  it  cannot  be   found  in   the  table,  taiic 
out  the  number  corresponding  to  the  next  less  logarithm   and 
set  it  aside.     Subtract  this  less  logarithm  from  the  given  lo^a 
rithm,   and  annex    to  the  remainder  as  many  zeros  as  may  bo 


INTRODUCTION.  J7l 

Of  LogarithmH. 

necessary,  and  divide  this  result  by  the  corresponding  numbei 
taken  from  the  column  marked  D,  continuing  the  division  as 
long  as  desirable  :  annex  the  quotient  to  the  number  set  aside. 
Point  ofl',  fiom  the  left  hand,  as  many  integer  figuies  as  there  are 
i.niUj  in  the  characteristic  of  the  given  logarithm  increased  by 
J  ;  tlie  result  is   the   required   number. 

If  the  characteristic  is  negative,  the  number  will  be  entirely 
decimal,  and  the  number  of  zeros  to  be  placed  immediately  after 
the  decimal  point  will  be  equal  to  the  number  of  units  in  the 
characteristic  diminished  by  1. 

This  rule,  like  its  converse,  is  founded  on  the  supposition  that 
the  difference  of  the  logarithms  is  proportional  to  the  difference 
of  their  numbers  within  narrow  limits. 

EXAMPLE. 

Find  the  number  corresponding  to  the  logarithm  3.233568. 

The  decimal  part  of  the  given  logarithm  is    .233568 

The  next  less  logarithm  of  the  table  is  .233504    and    its 

corresponding  number  1712.  

Their  difference  is         -  -  -  64 


Tabular  difference  253)6400000(25 

Hence  the  number  sought         1712.25 

The  number  corresponding  to  3.2335G8  is  .00171225 

MULTIPLICATION    BY    LOGARITHMS. 

10.  When  it  is  required  to  multiply  nujnbers  by  meanb  ot 
their  logarithms,  we  tirst  find  from  the  table  the  logarithms  of 
thfi  numbers  to  be  multiplied  ;  we  next  add  these  logarithms 
t<>g..  ther,  and  their  sum  is  the  logarithm  of  the  product  of  the 
numbers  (Art.  3). 

The   term    sum  is   to   be   understood  in   its  a'gebraic  seniw; 


172  TRIGONOMETRY. 


Of    Logarithms. 


therefore,  if  any  of  the  logarithms  have  negative  characteristicaaj 
the  diflference  between  their  sum  and  that  of  the  positive 
characteristics,  is  to  be  taken  ;  the  sign  of  the  remainder  u 
that  of  the  greater  sum. 

EXAMPLES. 

1.  Multiply  23.14  by  5.062. 

log  23.14  =  1.364363 
log  5.062  =  0.704322 


Product  117.1347  ....  2.068685 


2.  Multiply  3.902,  597.16  and  0.0314728  together. 

log    3.902  =  0.591287 

log  597.16  =  2.776091 

log  0.0314728  =  2.497936 

Product  73.3354  ....   1.865314 


Here  the  2  cancels  the  +  2,  and  the  1  carried  from  the  deci- 
mal part  is  set  down. 

8.  Multiply  3.586,  2.1046,  0.8372,  and  0.0294,  together, 
log     3.586  =  0.554610 
log  2.1046  =  0.323170 
log  0.8372  =  1.922829 
log  0.0294  =  2.468347 

Product    0.1857615     .     .     r.268956 


In  this  example  the  2,  carried  from  the  decimal  part^  cancols 
2,  and  there  remains  1  to  be  set  down. 

DIVISION    OF     NUMBERS     BY    LOGARITHMS. 

1 1 .  When  it  is  required  to  divide  numbers  by  means  of  tbeif 
lof^rithms,  we   have  only   to  recollect,   that  the   subtraction   of 


INTRODUCTION.  173 

Of    Logarithms. 

logarithms  corresponds  to  the  division  of  their  numbers  (Art.  3). 
Uence,  if  we  find  the  logarithm  of  the  dividend,  and  from  it  oub- 
tract  the  logarithm  of  the  divisor,  the  remainder  will  be  the  loga- 
rithm of  the  quotient. 

This  additional  caution  may  be  added.  The  diflference  of  tho 
logarithms,  as  here  used,  means  the  algebraic  difference;  so 
that,  if  the  logarithm  of  the  divisor  have  a  negative  characteristic 
its  sign  must  be  changed  to  positive,  after  diminishing  it  by  the 
unit,  if  any,  carried  in  the  subtraction  from  the  decimal  part  of 
the  logarithm.  Or,  if  the  characteristic  of  the  logarithm  of  tlie 
dividend  is  negative,  it  must  be  treated  as  a  negative  number. 

EXAMPLES. 

1.  To  divide  24163  by  4567. 

log  24163  =  4.383151 
W     4567  =  3.659631 


Quotient      5.29078     ....     0.723520 

2.  To  divide  0.06314  by  .007241 

log      0.0G314  =  2.800305 

log    0.007241  =  3.859799 

Quotient     .     .     8.7198    .     .     .     .     0.940506 

Here,  1  carried  from   the  decimal  part  to  the  3  changes  it  to 
2,  which  being  taken  from  2,  leaves  0  for  the  characteristic 

3    To  divide  37.149  by  523.76 

log  37.149  =  1.569947 
log  523.76  =  2.719133 


Quotient     .     .     0.0709274      .      2.850814 
15* 


174  TRIGONOMETRY. 

Of  Logarithm  B. 

4.  To  divide  0.7438  by  12.9476 

log     0.7438  =  1.871456 
log  12.9476  =  1.112189 

Quotient      .      .      0.057447      .      .     2".7592C7 

Here,  the  1  taken  from  1,  gives  2  for  a  result,  as  set  down. 


ARITHMETICAL    COMPLEMENT. 

12.  The  Arithmetical  complement  of  a  logarithm  is  the  num- 
ber which  remains  after  subtracting  the  logarithm  from  10. 
Thus,         .         .         1^—9.274687  =  0.725313 
Hence,  0.725313    is   the   arithmetical  complemem 

Df  9.274687. 


13.  We  will  now  show  that,  the  difference  betiveen  two  loga- 
rithms is  truly  found^  by  adding  to  the  first  logarithm  the 
arithmetical  comjdement  of  the  logarithm  to  be  subtracted^  and 
then  diminishing  the  sum  by  10. 

Let     a  =  the  first  logarithm 

b  =  the  logarithm  to  be  subtracted 
and  c  =  10  —  6  =  the  arithmetical  complement  of  6. 

Now  the  difference  between  the  two  logarithms  will  be  eX' 
pressed  by  a  —  b. 

But,  from  the  equation  c  =  10  — 6,  we  have 
c-10  =  —b 
hence,  if  wo   place  for— 6  its  value,  we  shall  have 

a—b  =  a-\-c—\0 
which  agrees  with   the  enunciation. 

When  we  wish  the  arithmetical  complement  of  a  logarithm, 
we  may  write  it  directly  from   the  table,  by  subtracting  the  le/\ 


INTRODUCTION.  176 

Of   Logarithms. 

hand  figure  from  9,  then  proceedinrj  to  the  riyht^  subtract  each 
Jifjure  from  9  till  we  reach  the  last  siynifcant  fiijure,  which 
must  be  taken  from.  10:  this  will  be  the  same  as  taking  th( 
logarithm  from  10. 

EXAMPLES. 

1.  From  3.274107     take     2.104729. 

By  common  method.  By  arith.  comp. 

3.274107  3.274107 

2.104729         its  ar.  comp.  7.895271 

Diff.      1.1G9378  Sum   1.1  G 93 7 8  after  sub- 


tracting 10. 

Ilenr^,  to  perform  division  by  means  of  the  aritlimetical  com* 
plenviiit  we  have  the  following 

RULE. 

To  the  logarithm  of  the  dividend  add  the  arithmetical  cofu- 
plemenf  of  the  logarithm  of  the  divisor :  the  sum  after  subtracts 
inq  in,  icill  be  the  logarithm  of  the  quotient. 

EXAMPLES. 

1.  Divide  32  7.5  by  22.07. 

log  327.5  .         .         .  2.515211 

log  22.07  ar.  comp.  8.65G198 

Quotient     .     .     14.839      ....     1.171409 

2.  Divide  0.7438  by  12.9476. 

log    0.7438 T.871456 

leg  12.9476         ar.  comp.         8.887811 

Quotient     .     .     0.057447      .      .      .      2.759267 


170  TRIGONOMETRY. 

Description    of   Instruments. 

In   this   example,    the  sum  of  the  characteristics  is  8,  from 
which,  taking  10,  the  remainder  is  2. 

3.  Divide  3Y.149  by  523.76. 

log  37.149 1.589947 

log  523.76         ar.  coup.  7.280867 

Quotient     .     .     0.0709273      .      .     .  2.850814 


SECTION   II. 


OF    SCALES. 


SCALE    OF    EaUAL    PARTS. 


t  , I    .1    a  ..T.4.5  -g  .7  .8  .97P 

J=  I  I     I      [     1     t      I      I     I     1—1— J. 

2,  J  a  \  if 

14.  A  scale  of  equal  parts  is  formed  by  dividing  a  lino  ol  a 
giver   length  into  equal  portions. 

Ii,  for  example,  the  line  ab  of  a  given  length,  say  one  inch, 
be  divided  into  any  number  of  equal  parts,  as  10,  the  scale  thus 
formed,  is  called  a  scale  of  ten  parts  to  the  inch.  The  line  a6, 
which  is  divided,  is  called  the  unit  of  the  scale.  This  unit  ib 
laid  off  several  times  on  the  left  of  the  divided  line,  and  its 
extremities  marked,  1,  2,  3,  <fec. 

The  unit  of  scales  of  equal  parts,  is,  in  general,  either  an 
inch,  or  an  exact  part  of  an  inch.     If,  for  example,  ah    the  unit 


INTRODUCTION. 


177 


Description    of   InBtraineuts. 


of  the  scale,  were  half  an  inch,  the  scale  would  be  one  of  10 
parts  to  half  an  inch,  or  of  20  parts  to  the  inch. 

If  it  were  required  to  take  from  the  scale  a  line  equal  to  two 
Inches  and  six-tenths,  place  one  foot  of  the  dividers  at  2  on  the 
'jcft,  and  extend  the  other  to  .6,  which  marks  the  sixth  of  the 
small  divisions :  the  dividers  will  then  embrace  the  required 
distance. 


DIAGONAL    SCALE    OF    EQUAL    PARTS. 


dj' 

c 

L                 1  / 

0» 

1             «    '  '  ' 

p 

08 

1   19^,        \   \ 

07 

,1       \    \       M 

06 

/   /       \    \\ 

I 

05 

\    4  \ 

04 

1              ' 

03 

1          \   \\\\\\ 

02 

\-   -U 

A-aA-X- 

01 

_i_± 

W^  TIT^ 

2 

/ 

i 

X 
1 

C 

\  ,1  .2  .3.4 

,5.G.7.^,S 

b 

16.  This  scale  is  thus  constructed.  Take  ah  for  the  unit  rf 
the  scale,  which  may  be  one  inch,  ^  |  or  J  of  an  inch,  in  length. 
On  ah  describe  the  square  ahcd.  Divide  the  sides  ah  and  dc 
each  into  ten  equal  parts.  Draw  af  and  the  other  nine  parallels 
as  in  the  figure. 

Produce  ha  to  the  left,  and  lay  off  the  unit  of  the  scale  any 
convenient  number  of  times,  and  mark  the  points  1,  2,  3,  ko.. 
Then,  divide  the  Hne  ad  into  ten  equal  parts,  and  through  the 
points  of  division  draw  parallels  to  ah  as  in  the  figure. 

Now,  the  small  divisions  of  the  line  ah  are  each  one-tenth 
(.1)  of  ah  ;  they  are  therefore  .1  of  cm/,  or  .1  of  ag  or  gh. 

If  we  consider  the  triangle  ad/,  we  see  that  the  base  df  is 


178  TRIGONOMEIRY 

Description    of   I  n  r  t  r  n  m  e  n  t  a. 

one-tenth  of  ad^  the  unit  of  the  scale.  Since  the  distance  from 
a  to  the  first  horizontal  line  above  a6,  is  one-tenth  of  the  dis- 
tance ad^  it  follows  that  the  distance  measured  on  that  hne  be- 
tween ad  and  af  is  one-tenth  of  df :  but  since  one-tenth  of  a 
tenth  is  a  hundredth,  it  follows  that  this  distance  is  one-hun- 
dredth (.01)  of  the  unit  of  the  scale.  A  like  distance  measured 
on  the  second  hne  will  be  two-hundredths  (.02)  of  the  unit  of 
the  scale ;  on  the   third,  .03 ;  on  the  fourth,  .04,  &c. 

If  it  were  required  to  take,  in  the  dividers,  the  unit  of  the 
scale,  and  any  number  of  tenths,  place  one  foot  of  the  dividers 
at  1,  and  extend  the  other  to  that  figure  between  a  and  6  which 
designates  the  tenths.  If  two  or  more  units  are  required,  the 
dividers  must  be  placed  on  a  point  of  division  further  to  the  left. 

When  units,  tenths,  and  hundredths,  are  required,  place  one 
foot  of  the  dividers  where  the  vertical  line  through  the  point 
which  designates  the  units,  intersects  the  hne  which  designates 
the  hundredths  :  then,  extend  the  dividers  to  that  hne  between 
ad  and  he  which  designates  the  tenths  :  the  distance  so  deter- 
mined will  be  the  one  required. 

For  example,  to  take  off  the  distance  2.34,  we  place  one  foot 
of  the  dividers  at  /,  and  extend  the  other  to  e :  and  to  take  oflf 
the  distance  2.58,  we  place  one  foot  of  the  dividers  at  p  and  ex- 
tend the  other  to  q. 

Remark  I.  If  a  line  is  so  long  that  the  whole  of  it  cannot 
be  taken  from  the  scale,  it  must  be  divided,  and  the  parts  of  it 
taken  from  the  scale  in  succession. 

Remark  II.  If  a  line  be  given  upon  the  paper,  its  length 
can  bo  found  by  taking  it  in  the  dividers  and  applying  it  to 
the  scale. 


INTRODUCTION 


179 


DeBoription    of  I  n  6  t  r  a  m  e  n  t  b. 


SCALE    OP    CHORDS 


»o  i'.9   7;q 

16.  if,  with  any  radius,  as  ^C,  we  describe  tlie  quadrant  (7Z>, 
and  then  divide  it  into  90  equal  parts,  each  part  is  called  a 
degree. 

Through  C,  and  each  point  of  division,  let  a  chord  be  drawn, 
and  let  the  lengths  of  these  chords  be  accurately  laid  otF  on  a 
Bcale :  such  a  scale  is  called  a  scale  of  chords.  In  the  figure, 
the  chords  are  drawn  for  every  ten  degrees. 

The  scale  of  chords  being  once  constructed,  the  radius  of  the 
circle  from  which  the  chords  were  obtained,  is  known ;  for,  the 
chord  marked  60  is  always  equal  to  the  radius  of  the  circle.  A 
scale  of  chords  is  generally  laid  down  on  the  scales  which  belong 
to  cases  of  mathematical  instruments,  and  is  marked  cho. 

To  lay  offj  at  a  given  point  of  a  line,  with  the  scale  nf  chords^ 
an  angle  equal  to  a  given  angle. 

Let  AB  be  the  line,  and  A   the  given 
point. 

Take  from  the  scale  the  chord  of  60  de- 
grees, and  with  this   radius,  and  the  point    A  S 
A  as  ti  centre,  describe  the  arc  BC.     Then  take  from  the  acale 


180 


TllIGONOMETRY 


DeBcription    of  Instruments. 


the  chord  of  the  given  angle,  say  30  degrees,  and  with  this  Hne 
as  a  radius,  and  ^  as  a  centre,  describe  an  arc  cutting  BC  m  C 
Through  A  and  C  draw  the  line  AC^  and  BAC  will  be  the  re- 
quired angle. 


SEMICIRCULAR  PROTRACTOR. 
C 


17.  This  instrument  is  used  to  lay  down,  or  protract  angles* 
It  may  also  be  used  to  measure  angles  included  between  lines 
already  drawn  upon  paper. 

It  consists  of  a  brass  semicircle  ABC  divided  to  half  degrees 
The  degrees  are  numbered  from  0  to  180,  both  ways;  that  is, 
from  ^  to  ^  and  from  B  to  A.  The  divisions,  in  the  figure, 
are  only  made  to  degrees.  There  is  a  small  notch  at  the  mid« 
die  of  the  diameter  AB^  which  indicates  the  centre  of  tie  pro- 
tractor. 

GUNTERS'    SCALE. 

18.  This  is  a  scale  of  two  feet  m  length,  on  the  faces  of 
which  a  variety  of  scales    is   marked.     The  face  on  which  the 


TRIGONOMETRY-  181 

Definitions. 

divisions  of  inches  are  made,  contains,  however,  all  the  scales 
necessary  for  laying  down  lines  and  angles.  These  are,  the 
scale  of  equal  parts,  the  diagonal  scale  of  equal  parts,  and  the 
EcaJe  of  chords,  all  of  which  have  been  described. 


PLANE    TRIGONOMETRY. 


DEFINITIONS    AND    EXPLANATION    OF    TABLES. 


19.  In  every  plane  triangle  there  are  six  parts:  three  sides 
and  three  angles.  These  parts  are  so  related  to  each  other,  that 
when  one  side  and  any  two  other  parts  are  given,  the  remain- 
ing parts  can  be  obtained,  either  by  geometrical  construction  or 
by  trigonometrical  computation. 

20.  Plane  Trigonometry  explains  the  methods  of  computing 
the  unknown  parts  of  a  plane  triangle,  when  a  sufficient  num- 
ber of  the  six  parts  is  given. 

21.  For  the  purpose  of  trigonometrical  calculation,  the  cir- 
cumference of  the  circle  is  supposed  to  be  divided  into  360 
ftqual  parts,  called  degrees ;  each  degree  is  supposed  to  be  di- 
fided  into  60  equal  parts,  called  minutes  ;  and  each  minute  into 
00  equal  parts,  called  seconds. 

Degrees,   minutes,   and   seconds,   are  designated  respectively 
IB 


IK2 


TRIGONOMETRY 


Definitions. 


by  the  characters  °  '  ".  For  example,  ten  degrees^  eighteen 
minutes^  and  fourteen  seconds^  would  be  written  10°  18'  14" 
If  two  lines  be  drawn  through  the  centre  of  the  circle,  al 
right  angles  to  each  other,  they  will  divide  the  circumference 
intc  four  equal  parts,  of  90°  each.  Every  right  angle  then,  as 
EOA,  is  measured  by  an  arc  of  90° ;  every  acute  angle,  as 
BOA,  by  an  arc  less  than  90°  ;  and  every  obtuse  angle,  as 
FOA,  by  an  arc  greater   than  90°. 

22.  The  complement  of  an  arc  is 
what  remains  after  subtracting  the 
arc  from  90°.  Thus,  the  arc  UB  is 
the  complement  of  AB.  The  sum  of 
an  arc  and  its  complement  is  equal 
to  90°. 

23.  The    suirplement    of  an   arc    is 
what   remains    after   subtracting    the 
arc  from  180°.     Thus,  6^i^  is  the  sup- 
plement of  the  arc  AEF.     The   sum  of  an   arc  and   its   sup- 
plement is  equal  to  180°. 

24.  The  sine  of  an  arc  is  the  perpendicular  let  fall  from  one 
extremity  of  the  arc  on  the  diameter  which  passes  through 
the  other  extremity.     Thus,  BD  is   the   sine  of  the  arc  AB. 

25.  The  cosine  of  an  arc  is  the  part  of  the  diameter  inter- 
cepted between  the  foot  of  the  sine  and  centre.  Thus,  OD  is 
the  cosine  of  the  arc  AB. 

26.  The  tangent  of  an  arc  is  the  line  which  touches  it  at 
one  extremity,  and  is  limited  by  a  line  drawn  through  the 
other  extremity  and  the  centre  of  the  circle.  Thus,  AQ  \%  the 
Ungent  of  the  arc  AB. 


TRIGONOMETRY 


183 


Definitions. 


27.  The  secant  of  an  arc  is  the  hne  drawn  from  the  ceDti^ 
of  the  circle  through  one  extremity  of  the  arc,  and  limited  by 
the  tangent  passing  through  the  other  extremity.  Thus,  0(' 
-s  the  secant  of  the  arc  AB, 


28.  The  four  Hnes,    BD,    OD,   AG,    OC,  depend   for  tlicir 
Nalacs  on    the  arc   AB   and   the  radius    OA;  they   are   thus 


designated : 

sin  AB 

for 

BJ) 

cos  AB 

for 

OD 

tan  AB 

for 

AC 

sec  AB 

for 

OC 

n 

V 

B 

tA 

\ 

T^X, 

G 

r 

\ 

1 

\ 

f       0 

X 

29.  If  ABE  be  equal  to  a  quad- 
rant, or  90°,  then  EB  will  be  the 
complement  of  AB.  Let  the  lines 
ET  and  IB  be  drawn  perpendicular 
to   OE.     Then, 

ET,  the  tangent  of  EB,  is  called  the  cotangent  of  AB  : 
IB,  the  sine  o^  EB,  is  equal  to  the  cosine  of  AB ; 
OT,  the  secant  of  EB,  is  called  the  cosecant  of  AB. 
In  general,  if  A  is  any  arc  or  angle,  we  have, 
cos  A  —  sin  (90°  —  ^) 
cot  A  =  tan  (90°  —  ^) 
cosec  A  =  sec  (90^  —  ^) 

30.  If  we  take  an  arc  ABEF,  greater  than  90°,  its  sme 
will  be  FH ;  0^  will  be  its  cosine;  ^^  its  tangent,  and  OQ 
its  secant.  But  FH  is  the  sine  of  the  arc  GF,  which  is  the 
supplement   of  AF,   and  OE  is  its   cosine  :  hence,  the  sine  of 


184  TRIGONOMETRY. 

Definitions. 

an  arc  is  equal  to  the  sine  of  its  supplement;  and  the  cosiru 
of  an  arc  is  equal  to  the  cosine  of  its  supplement* 

Furthermore,  ^  ^  is  the  tangent  of  the  arc  AF,  and  OQ  it 
its  secant:  GL  is  the  tangent,  and  OL  the  secant  of  the  sup- 
plemental arc  GF.  But  since  ^^  is  equal  to  GL,  and  OQ 
to  OL,  it  follows  that,  the  tangent  of  an  arc  is  equal  to  the 
tannent  of  its  supplement;  and  the  secant  of  an  arc  is  equal 
to  the  secant  of  its  supplement.* 

Let  us  suppose,  that  in  a  circle  of  a  given  radius,  the 
lengths  of  the  sine,  cosine,  tangent,  and  cotangent,  have  been 
calculated  for  evefy  minute  or  second  of  the  quadrant,  and 
arranged  in  a  table ;  such  a  table  is  called  a  table  of  sines  and 
tangents.  If  the  radius  of  the  circle  is  1,  the  table  is  called  a 
table  of  natural  sines.  A  table  of  natural  sines,  therefore,  shows 
the  values  of  the  sines,  cosines,  tangents  and  cotangents  of  all 
the  arcs  of  a  quadrant,  divided  to  minutes  or  seconds. 

If  the  sines,  cosines,  tangents,  and  secants  are  known  for  area 
less  than  90°,  those  for  arcs  which  are  greater  can  be  found 
from  them.  For  if  an  arc  is  less  than  90°,  its  supplement 
will  be  greater  than  90°,  and  the  values  of  these  lines  are  the 
same  for  an  arc  and  its  supplement.  Thus,  if  we  know  the 
sine  of  20°,  we  also  know  the  sine  of  its  supplement  160^  ; 
for  the  two  are  equal  to  each  other. 

TABLE    OF    LOGARITHMIC    SINES. 

31.  In  this  table  are  arranged  the  logarithms  of  the  nume- 
rical values  of  the  sines,  cosines,  tangents,  and  cotangents  of  all 

*  These  relations  are  between  the  numerical  values  of  the  trigonometrica] 
lines;  the  algebraic  signs,  which  thoy  have  in  the  diifcrent  quadrants,  art? 
Dot  considered. 


TRIGONOMETRY  185 

Uses  of  tho  ToblcB. 

the  arcs  of  a  quadrant,  calculated  to  a  radius  of  10,000,000,000. 
The  logarithm  of  this  radius  is  10.  Iii  the  first  and  last  hori- 
zontal lines  of  each  page,  are  written  the  degrees  whose  sines, 
cosines,  <fec,  are  expressed  on  the  page.  The  vertical  coiumna 
on  the  left  and  right,  are  columns  of  minutes. 

CASE    I. 

To  find,  in  tlte  table,  the  logarithmic   sine,  cosine,  tangent,  or 
cotangent  of  any  given  arc  or  angle. 

82.  If  the  angle  is  less  than  45°,  look  for  the  degrees  in  the 
first  horizontal  line  of  the  different  pages :  then  descend  along 
the  column  of  minutes,  on  the  left  of  the  page,  till  you  reach 
the  number  showing  the  minutes :  then  pass  along  the  hori- 
zontal line  till  you  come  into  the  column  designated,  sine, 
cosine,  tangent,  or  cotangent,  as  the  case  may  be :  the  numbei 
BO  indicated  is  the  logarithm  sought.  Thus,  on  page  37,  foi 
19^  65'  we  find, 

sine  19°  55' 9.532312 

cos  19°  55' 9.973215 

tan  19°  55' 9.559097 

cot  19°  55' 10.440903 

33.  If  the  angle  is  greater  than  45°,  search  for  the  degrecb 
along  the  bottom  line  of  the  different  pages :  then,  ascend 
Along  the  column  of  minutes  on  the  right  hand  side  of  the 
page,  till  you  reach  the  number  expressing  the  minutes :  then 
pass  along  the  horizontal  Une  into  the  column  designated 
tang  cot,  sine,  or  cosine,  as  the  case  may  be:  the  number  p<i 
pointed  out  is  the  logarithm  required. 

34.  The  column  designated  sine,   at  the  top  of  the  page,  is 

10* 


iW«  TRIGONOMETRY. 


Uses  of  the  Tables. 


designated  by  cosine  at  the  bottom  ;  the  one  designated  tang, 
by  cotang,  and  the  one  designated  cotang,  by  tang. 

The  angle  found  by  taking  the  degrees  at  the  top  of  the 
page  and  the  minutes  from  the  first  vertical  column  on  th^ 
l3ft,  is  the  complement  of  the  angle  found  by  taking  the  de- 
grees at  the  bottom  of  the  page,  and  the  minutes  traced  up  in 
the  right  hand  column  to  the  same  horizontal  line.  There- 
fore, sine,  at  the  top  of  the  page,  should  correspond  with  cosine, 
at  the  bottom  ;  cosine  with  sine,  tang  with  cotang,  and  cotang 
with  tang,  as  in  the  tables  (Art.   11). 

If  the  angle  is  greater  than  90°,  we  have  only  to  subtract  it 
from  180°,  and  take  the  sine,  cosine,  tangent  or  cotangent  of 
the  remainder. 

ITie  column  of  the  table  next  to  the  column  of  sines,  and 
on  the  right  of  it,  is  designated  by  the  lette'*  D,  This  column 
is  calculated  in  the  following  manner. 

Opening  the  table  at  any  page,  as  42,  the  sine  of  24°  is 
found  to  be  9  609313;  that  of  24°  01',  9.609597:  their  dif- 
ference is  284  ;  this  being  divided  by  60,  the  number  of  seconds 
in  a   minute,  gives  4.73,  which  is  entered  in  the  column  D. 

Now,  supposing  the  increase  of  the  logarithmic  sine  to  be 
proportional  to  the  increase  of  the  arc,  and  it  is  nearly  so  for 
60",  it  follows,  that  4.73  is  the  increase  of  the  sine  for  1". 
Similarly,  if  the  arc  were  24°  20'  the  increase  of  the  sine  for 
1",  would  be  4.65. 

The  same  remarks  are  applicable  in  respect  of  the  column 
i>,  after  the  column  cosine,  and  of  the  column  i>,  between 
tho  tangents  and  cotangents.  The  column  D  between  the 
columns  tangents  and  cotangents,  answers  to  both  of  thcst 
columns. 


TRIGONOMETRY.  197 

Ubob  of  the  Tables. 

Now,  if  it  were  required  to  find  the  logarithmic  sine  of  an 
arc  expressed  in  degrees,  minutes,  and  seconds,  we  have  only 
to  find  the  degrees  and  minutes  as  before;  then,  multiply  tJie 
corresponding  tabular  difference  by  the  seconds,  and  add  the  pro 
duct  to  the  number  first  found,  for  the  sine  of  the  given  arc 

Thus,  if  we  wish  the  sine  of  40°  26'  28". 

The  sine  40°  26' 9.811952 

Tabular  difft-rence         2.47  . 

Number  of  seconds         28  . 


Product     .     .     69.16  to  be  added  69.10 

Gives  for  tlie  sine  of  40°  26'  28"  9.812021. 


The  decimal  figures  at  the  right  are  generally  omitted  in 
the  final  result ;  but  when  they  exceed  five-tenths,  the  figure  on 
tlie  left  of  the  decimal  point  is  increased  by  1  ;  this  gives  the 
nearest  approximate  result. 

The  tangent  of  an  arc,  in  which  there  are  seconds,  is  found 
in  a  manner  entirely  similar.  In  regard  to  the  cosine  and  co- 
tangent, it  must  be  remembered,  that  they  increase  while  the 
arcs  decrease,  and  decrease  as  the  area  are  increased ;  conse- 
quently, the  proportional  numbers  found  for  the  seconds,  musl 
be  subtracted,  not  added. 

EXAMPLES. 

1.  To  find  the  cosine  of  3°  40'  40" 

The  cosine  of  3°  40'  ...         9.999110 

Tabular  difference         .13     . 
Number  of  seconds         40   . 

Product  6.20  to  be  subtracted 6^ 

Gives  for  the  cosine  of  3°  40'  40"       .         9.999105 


188  TRIGONOMETRY. 


T)  BCB  of  the  TabloB. 


2.  Find  the  tangent  of  37°  28'  31" 

Ans.  9.884692. 
8.  Find  the  cotangent  of  87°  57'  59" 

Ans.  8.550356. 

CASE    II. 

To  find  the  degrees,  minutes  and  seconds^  answering  to  any 
given  logarithmic  sine,  cosine,  tangent  or  cotangent. 

35.  Search  in  the  table,  and  in  the  proper  column,  and  if  the 
number  be  found,  the  deo^rees  will  be  shown  either  at  the  top 
or  bottom  o^"  the  page,  and  the  minutes  in  the  side  columns, 
either  at  the  left  or  right. 

But,  if  the  number  cannot  be  found  in  the  table,  take 
from  the  table  the  degrees  and  minutes  answering  to  the  near- 
est less  logarithm,  the  logarithm  itself,  and  also  the  corres- 
ponding tabular  difference.  Subtract  the  logarithm  taken  from 
the  table  from  the  given  logarithm,  annex  two  ciphers  to  the 
remainder,  and  then  divide  the  remainder  by  the  tabular  dif- 
ference :  the  quotient  will  be  seconds,  and  is  to  be  connected 
with  the  degrees  and  minutes  before  found ;  to  be  added  for 
che  sine  and  tangent,  and  subtracted  for  the  cosine  and  co- 
tangent. 

EXAMPLES, 

1.  Find  the  arc  answering  to  the  sine         9.880054 
Sine  49°  20',  next  less  in  the  table         9.879963 

Tabular  difference     .         .         .         1.81)91.00(50" 
Hence,   the  arc  49°  20'  50"  corresponds   to  the  given  sid*: 
9  880054. 

2.  Find  the  arc  whose  cotangent  is      .     10.008688 
cot  44°  26',  next  less  in  the  table     .      10.008591 


Tabular  difference    .         .         .  4.21)97.00(28" 


TRIGONOMETRY.  189 

Theoroms. 

Hence,  44°  26'-23"  =  44°  25'  37'' is  the  arc  answering  to 
the  given  cotangent  10.008688. 

8.  Find  the  arc  answering  to  tangent  9.979110, 

Ans.  43°  37'  21". 

4    Find  the  arc  answering  to  cosine  9.944599. 

Ans.  28°  19'  46". 

86.   We    shall  now   demonstrate  the    principal    theorems  of 
Plane  Trigonometry. 

THEOREM    I. 

The  sides   of  a  plane    triangle   are  proportional    to    the  sines 
of  their  opposite  angles. 

Let   ABC  be    a   triangle;  then   will 

CB  :  CA  ::  sin  A  :  sin  B. 
For,    with  ^^  as  a  centre,  and  AD 
equal  to  the  less  side -5(7,  as  a  radius, 
describe  the  arc  DI:  and  with   B  as 
a   centre   and   the    equal    radius    BC^     ^         Kl  L 
describe  the    arc  CL'.  now  BE  is    the    sine   of  the   angle  A^ 
and   CF  is  the  sine  of  B^  to  the   same  radius   AD  ov  BC. 
But  by  similar  triangles, 

AD  \  DE  \\  AC  \   CF. 
But  AD  being  equal  to  BCj  we  have 

BC  :  sin  ^  :  :  ^C  :  sin  B,  or 
BC  :  AC  :  :  sm  A  '.  sin  B. 
By  comparing  the  sides  AB    ^C,  in  a  similar  manner,  we 
should  find,         AB  -.   AC  \  '.  ^m   C  :  sin   B. 


190  TRIGONOMETRY. 

Theorems. 

THEOREM    II. 

In  any  triangle,  the  sum  of  the  two  sides  containing  eithei 
angle,  is  to  their  diference,  as  the  tangent  of  half  the  sum  of 
the  two  other  angles,  to  the  tangent  of  half  their  differerxc. 

IjQt  AC B  be  a  triangle:  then  will 

AB-\-AC:AB--AC:'.i2in\{C-{-B)  :  tan  i(C-5). 

With  ^  as  a  centre,  and  a  radius 
AC  the  less  of  the  two  given  sides, 
let  the  semicircle  IFCE  be  de- 
scribed, meeting  AB  in  /,  and  BA 
produced,  in  E.  Then,  BE  will 
be  the  sum  of  the  sides,  and  BI 
Hieir  difference.     Draw  CI  ^w<S.  AF. 

Since  CAE  is  an  outward  angle  of  the  triangle  ACB.  it 
is  equal  to  the  sum  of  the  inward  angles  G  and  B  (lik. 
I,  Th.  xvi.)  But  the  angle  CIE  being  at  the  circumference, 
is  half  the  angle  CAE  at  the  centre  (Bk.  II,  Th.  viii.  Cor- 
1) ;  that  is,  half  the  sura  of  the  angles  C  and  B,  or  equal 
to|(C+^). 

The  angle  AFC  =  ACB,  is  also  equal  to  ABC  -{-  BAF ; 
therefore,  BAF  =  ACB  -  ABC. 

But,  ICF=  ^{BAF)  =  ^{ACB  —  ABC),  or  1{C—B). 

With  /  and  C  as  centres,  and  the  common  radius  IC,  let 
the  arcs  CD  and  IG  be  described,  and  draw  the  lines  CE  and 
IH  perpendicular  to  IC.  The  perpendicular  CE  will  pasa 
through  E,  the  extremity  of  the  diameter  IE,  since  the  right 
angle  ICE  must  be  inscribed  in  a  semicircle. 

But  CE  is  the  tangent  of  CIE  =  U^-^B) ;  and  Iff  is  tlie 
tangent  of  ICB  =  ^[C — B),  to  the  common  radius   CI. 


TRIGONOMETRY. 


191 


TheoromH. 


But   since   tlie   lines    CE  and  IH  are  parallel,  the  trianglee 
BHI  and  BCE  are  similar,  and  give  the  proportion, 

BE  '.  BI  '.:   CE  :  III,  or 

by  placing   for  BE  and  BI,  CE  and  III,  their  values,  we  have 

AB  ^  AC  \  AB  —  AC  w  tan  \{C-VB)  :  tan  \{C  —  B\ 


THEOREM    IIL 

In  any  plane  triangle,  if  a  line  is  drawn  from  the  vertical 
angle  perpendicular  to  the  base,  dividing  it  into  two  segments: 
tJien,  the  whole  base,  or  sum  of  the  segments,  is  to  the  sum  of 
the  two  other  sides,  as  the  difference  of  those  sides  to  tJie  dif- 
ference of  the  segments. 

Let  BAC  be  a  triangle,  and  AD  perpendicular  to  the  base; 
then  will 

BC  :  CA  +  AB  :\  CA  —  AB  :  CD  —  DB 

For,  ~Ai? =inj  ^~nf 

(Bk.  IV,  Th.  xii) ; 

and  ~AG'  =  DG"  +  ~Alf 

by  subtraction  .4(7*  —  TS  =  C'X>  — 

Ti^. 

But  since  the  diflference  of  the  squares 
of  two  lines  is  equivalent  to  the  rectangle  contained  by  their  sum 
and  diflference  (Davies*  Legendre,  Bk.  IV,  Prop,  x,)  we  havo, 

AC"  —  A^  =  {AC -\- AB)  .  {AC—AB) 
ind  'C^  —  'dF=  {CD  +DB).{CD  —  DB) 

therefore,  (CD  +  DB) .  (CD  —  DB)  =  (AC-]'AB).(AC-A/i] 
hence,        CD -^DB  :  AC  +  AB :  :AC—AB:CD  —  DB. 


192 


TRIGONOMETRY 


Theorems. 


THEOREM    IV. 

In  any  right-angled  plane  triangle,  radius  is  to  the  tan^ 
gent  of  either  of  the  acute  angles,  as  the  side  adjacent  to  thi 
ktde  opposite. 

Let    CAB    be    the    proposed    triangle,  j^ 

and  denote  the  radius  by  B  :  then  will 
Ji  :  tan  C::AC  :  AB. 
For,   with  any   radius   as    CD  describe  ^^ 
the  arc  DH,  and  draw  the  tangent  DG. 

From  the  similar  triangles    CDG  and  CAB  we  have 
CD  :DG  ::  CA:  AB',  hence, 
E:  tan  C  ::  CA:  AB. 
By  describing  an  arc  with  i?  as   a  centre,  we   could  show    in 
the  same  manner  that, 

B  '  tan  B  ::AB  :  AC. 


THEOREM    V. 

In  every  right-angled  plane  triangle,  radius  is  to  the  cosine 
of  either  of  the  acute  angles,  as  the  hypothenuse  to  iJie  side 
adjacent. 

Let  ABC    be    a   triangle,  right-angled 
at  B  then  will 

R  :  cos  A'.'.AC  :  AB.  Ur 

For,  from  the  point  yl  as  a  centre,  with  y. 
nny  radius  as   AD,  describe  the  arc  DF, 
which  will  measure  the   angle  A,  and  draw  DE  perpendlculai 
to  AB  :  then  will  AE  be  the  cosine  of  A. 

The  triangles  ADE  and  ACB,  being  similar,  we  have 
AD  '.AE  '.'.AC  '.AB:  that  is, 
E  :  oos  A  ::  AC  :  AB. 


TRIGONOMETRY.  193 

Application  B. 

Remark.  The  relations  between  the  sides  and  angles  of 
plane  triangles,  demonstrated  in  these  five  theorems,  are  suf- 
ficient to  solve  all  the  cases  of  Plane  Trigonometry.  Of  the 
i\\  parts  which  make  up  a  plane  triangle,  three  must  be  given, 
and  at  least  one  of  these  a  side,  before  the  others  can  be  de- 
termined. 

If  the  three  angles  are  given,  it  is  plain,  that  an  indefi- 
nite number  of  similar  triangles  may  be  constructed,  the 
angles  of  which  shall  be  respectively  equal  to  the  angles 
that  are  given,  and  therefore,  the  sides  could  not  be  de- 
termined. 

Assuming,  with  this  restriction,  any  three  parts  of  a  trian* 
gle  as  given,  one  of  the  four  following  cases  will  always  be  pre 
sonled. 

I.  When  two  angles  and  a  side  are  given. 
II.  When  two  sides  and  an  opposite  angle  are  given, 

III.  When  two  sides  and  the  included  angle  are  given, 

IV.  When  the  three  sides  are  given. 

CASE    I. 

When  two  angles  and  a  side  are  given. 

Add  the  given  angles  together  and  subtract  their  sum  from 
ISO  degrees.  The  remaining  parts  of  the  triangle  can  then 
be  found  by  Theorem  I. 

KXAMPLES. 

1.  In   a   plane  triangle   ABCj    there 
are  given   the  ang'e  A  =  58°  07',  the 
angle  B  ^  22°  37  ,  and  the  side  AB  =   A 
i08  J  ards.     Required  the  other  parts. 
17 


194  TRIGONOMETRY. 

Application  B. 

GEOMETRICALLY. 

Draw  an  indefinite  straight  line  AB^  and  from  the  d'^ale  ct 
ecjual  parts  lay  off  AB  equal  to  408.  Then  at  A  lay  off  aE 
angle  equal  to  58°  07',  and  at  B  an  angle  equal  to  ?2°  37', 
and  draw  the  lines  AC  and  BC :  then  will  ABC  be  the  tri- 
angle required. 

The  angle  C  may  be  measured  either  with  the  protractor  or 
the  scale  of  chords  (Arts.  16  and  17),  and  will  be  found  equal 
to  99°  16'.  The  sides  AC  and  BC  may  be  measured  by  re- 
ferring them  to  the  scale  of  equal  parts  (Art.  2).  W«  *hall 
6nd  AC  =  158.9  and  BC  =  351.  yards. 


TRIuOyOMfiTKICALLT    BY    LOGARITHMS. 

To  the  angle     . 

.     ^  =  58°  07' 

Add  the  angle 

.     B  =  22°  37' 

Their  sum 

=  80°  44' 

taken  from  .     . 

180°  00' 

leaves  C       .     . 

99°  16'  which. 

exceeding  00" 

we  use  its  supplement 

80°  44'. 

To 

find  the  side  BC. 

As  sin   C       99°  16' 

ar.  comp.    . 

0.005705 

:  sm      A        58°  07' 

; 

9.928972 

:  :      AB           408 

.... 

2.610660 

;           BC      361.024 

(after  rejecting  10) 

2.545337 

Remark.  The  logarithm  of  the  fourth  term  of  a  propoition 
IS  obtained  by  adding  the  logarithm  of  the  second  term  to  th.il 
of  the  third,  and  subtracting  from  their  sum  the  logarithm  of 
the   first  term.     But  to   subtract  the  first  term  is  the  same  ae 


TRIOONOMETRT 


195 


Application  B. 


to  .'idd  its  arithmetical  complemen*  and  reject  IC  from  the  sum 
(Art.  13)  :  hence,  the  arithmetical  complement  of  the  first 
term  added  to  the  logarithms  of  the  second  and  third  terms, 
micus  ten,  will  give  the  logarithm  of  the  fourth  term. 

To  find  side  AC, 


As   sin    C 

99°  16' 

:  sin       B 

22°  37' 

:  :      AB 

408 

:          AC 

158.976 

ar.  comp. 


0.005705 
9.584968 
2.610660 
2.201333 


2.  In  A  triangle  ABC^  there  are  given  A  =  38°  25', 
B  =  67°  42',  and  AB  =  400 :  required  the  remaining  parts. 
Ans.  C  =  83°  53',  BC  =  249.974,  AC  =  34C.04 


CASE    II. 

When  two  sides  and  an  opposite  angle  are  given. 
In  a  plane  triangle  ABC^  there  are 
given    AC  =   216,     C^  =  117,    the 
angle  ^  =  22^*  37',   to   find  the  other 
paita. 


GEOMETRIC  ALLY. 

Draw  an  mdefinite  right  line  ABB' :  from  any  point  as  Aj 
draw  AC  making  BAC  =  22°  37',  and  make  AC  =  216. 
With  0  as  a  centre,  and  a  radius  equal  to  117,  the  other  given 
side,  describe  the  arc  B'B;  draw  B' C  and  BC:  then  will 
cither  of  the  triangles  ABC  ot  AB  Cy  answer  all  the  condi- 
tiona  of  the  question. 


196  TRIGONOMETRY. 


Applicati  ons. 


TRIGONOMETRIC  ALLY. 

To  find  the  angle  B, 

As    5(7      117             .       ar.  comp.     .        .  7.931814 

',        AC     2\Q              2.334454 

;  :  sin  ^      22°  37' 9.584968 

:     sin  B'  45°  13'  55",  or  ABC  134°  4G'  05"  9.851236 

The  ambiguity  in  this,  and  similar  examples,  arises  in  con- 
sequence of  the  first  proportion  being  true  for  either  of  the 
angles  ABC^  or  AB'Cj  which  are  supplements  of  each  other, 
and  therefore  have  the  same  sine  (Art.  30).  As  long  as  the 
two  triangles  exist,  the  ambiguity  will  continue.  But  if  the 
side  CB,  opposite  the  given  angle,  is  greater  than  AC,  the  arc 
BB'  will  cut  the  hne  ABB',  on  the  same  side  of  the  point  A, 
in  but  one  point,  and  then  there  will  be  only  one  triangle  an- 
Bv/ering  the  conditions. 

If  the  side  CB  is  equal  to  the  perpendicular  Cd,  the  arc 
BB'  will  be  tangent  to  ABB',  and  in  this  case  also  there 
will  be  but  one  triangle.  When  CB  is  less  than  the  perpen- 
dicular Cd,  the  arc  BB'  will  not  intersect  the  base  ABB',  and 
m  that  case,  no  triangle  can  be  formed,  or  it  will  be  impossible 
to  fulfil  the  conditions  of  the  problem. 

2.  Given  two  sides  of  a  triangle  50  and  40  respectively,  and 
the  angle  opposite  the  latter  equal  to  32°  :  required  the  re- 
maining parts  of  the  triangle. 

Ans.  If  the  angle  opposite  the  side  50  is  acute,  it  is  equal 
U)  41°  28'  59"  ;  the  third  angle  is  then  equal  to  106°  31'  01", 
and  the  third  side  to  72.368.     If  the  angle  opposite  the  side 


TRIGONOMETRY.  19T 


Applications. 


50  is  obtuse,  it  is  equal  to  138°  31'  01",  the  third  angle  to 
9**  28'  69",  and  the  remaining  side  to  12.436. 

CASE    III. 

When  the  two  sides  and  their  included  angle  are   given. 

Let  ABC  he  a  triangle;  AB,  BC, 
the  given  sides,  and  B  tbe  given 
angle. 

Since  B  is  known,  we  can  find  the 
sura  of  the  two  other  angles :  for 

A-\-  C  =  180°  —  B  and 
\{A  +  C)  =  J(180°  -  B) 
We  next  find  half  the  diflfereuce  of  the  angles  A  and  C  by 
Theorem  ii.,  viz. 

BC  +  BA'.BC-BA\'.  tan  \{A  +  C)  :  tan  i(A  -  C): 
in  which  we  consider  BO  greater  than  BA,  and  therefore  A  ia 
greater  than  C;  since  the  greater  angle  must  be  opposite  the 
greater  side. 

Having  found  half  the  difference  of  A  and  0,  by  adding  it 
to  the  half  sum,  ^(A  -f  C),  we  obtain  tlie  greater  angle,  and  by 
subtracting  it  from  half  the  sum,  we  obtain  the  less.     That  ia 
1{A  +  C)  H-  l(A  -C)  =  A,  and 
i{A-{-C)-i(A-.C)=  C. 
Having  found  the  angles  A  and  (7,  the  third  side  AO  may 
be  found  by  the  proportion. 

sin  A  :  sin  B  :  :  BC  :  AC. 

EXAMPLES. 

1.  In  the  triangle  ABC,  let  BC  =  540,  AB  =  450,  and 
the  included  angle  B  =  80°  :  required  the  remaining  parts. 
17* 


198  TRIGONOMETRY 

Applications. 


GEOMETRICALLY. 


Draw  an  indefinite  right  line  BC  and  from  any  point,  as 
/?,  lay  oflf  a  distance  BC  =  540.  At  B  make  the  angle 
CBA  —  80°  :  draw  BA  and  make  the  distance  B  A  --  450 1 
draw  AC\  then  will  ABC  be  the  required  triangle. 


TRIGONOMETRICALLY. 


BC  +  BA  z=  540  +  450  =  990 ;  and  BG  —  BA  =  640  — 

450  =  90. 

ui  +  e  =  ISO*'  —  ^  =  180°  —  80°  =  100^  and  therefore, 

i(^4-  (7)=i(100°)  =60* 


\0 


To  find  \[A—C). 
As  BC  -\-  BA  990       .      ar.  comp.      .     7.004366 

BC—BA  90       .         .         .  .     1.954243 

tan  ^{A  +  (7)  60°      .         .         .  .  10.076187  ^ 

tan  ^(A—C)  6°  11'         .         .  .     9.034795 

Ifence,  50°  +  6°  11'  =  56°   11'  =  A-  and  60°  —  6°  11'  = 
43°  49'  =  C. 


To  find  the  third  side  AC, 

As  sin  C 

43°  49'     .     ar.  comp.     . 

.     0.159672 

:     sin  B 

80°         ...         . 

.     9.993361 

::        AB 

450           .... 

.     2.653213 

AC 

640.C82    .... 

.     2.80623C 

2.  Given   two  sides  of  a  plane  triangle,  1686  and  960,  and 
iheir  included  angle  128°  04' :  required  the  other  parts. 

Ans.  Angles,  33°  34'  39";  18°  21'  21";  side  2400. 


TRiGONOMEl   RY. 


199 


Applications. 


CASE   IV. 

Having  given  the  three  sides  of  a  plane  triangle,  to  find 
die  angles. 

Let  tali  a  perpendicular  from  the  angle  opposite  the  greater 
tide,  dividing  the  given  triangle  into  two  right-angled  triangles  : 
then  find  the  diflerence  of  the  segments  of  the  bfise  by  Theo- 
rem iii.  Half  this  diflerence  being  added  to  half  the  base^ 
gives  the  greater  segment ;  and,  being  subtracted  from  half  the 
base,  gives  the  less  segment.  Then,  since  the  greater  segment 
belongs  to  the  right-angled  triangle  having  the  greatest  hypo- 
thenuse,  we  have  the  sides  and  right  angle  of  two  right-angled 
triangles,   to  find  the  acute   angles. 


EXAMPLES. 

1.  The  sides  of  a  plane  trian- 
gle being  given;  viz.  BC  =  40,-4(7 
=  34  and  AB  =  26  :  required  the 
angles.  B 

GEOMETRICALLY, 

With  the  three  given  lines  as  sides  construct  a  triangle  as 
in  Bk.  11.  Prob.  xi.  Then  measure  the  angles  of  the  triangle 
either  with  the  protractor  or  scale  of  chords. 

TRIGONOMETRIC  ALLY. 

As  BC  :  AC  -\-  AB  :  :  AC  -  AB  :  CD  -  BD 

Tliat  is,  40      :      59      :  :      9      :      ^^  ^  ^  =  13.27c 

40 
40  +  13.275 


Then, 
And 


=  26.6375 


40 
CD 


40  —  13.275 


13.3625  =  BD. 


200  TRIGONOMETRY. 

ApplicationB. 

In  the  triangle  J) AC,  to  find  the  angle  DAC, 

Ab           AC           34     .         .        ar.  comp.     ,  8.468521 

DC          26.6375     ....  1.42549S 

sin   i>              90° 10.000000 

sin   DAC       51'>  34'  40"       .         .         .  9.8940 N 

In  the  triangle  BAD,  to  find   the  angle  BAD. 
As  AB  25  ar.  comp.         .         8.602060 

BD  13.3625  .         .         .         1.125887 

sin   D  90°  ...         .       10.000000 

sin   BAD        32°  18'  35"    .         .         .         9.727947 
Hence  00°  — i>^(7  =  90°  —  51°  34' 40"  =  S8°  25' 20"  =  C 
and       90°  —  BAD  =  90°  —  32°  18'  35"  =  57°  41'  25"  =  B 
and  BAD  +  DAC=  51°  34'  40"  +  32°  18' 35"  =  83°  53' 
15"  =  A. 

2.  In  a  triangle,  in  which  the  sides  are  4,  5  and  6,  what  are 
the  angles  ? 

Ars.  41°  24'  35";  55°  46'  16";  and  82°  49'  09". 

SOLUTION     OF     RIGHT-ANGLED     TRIANGLES. 

The  unknown  parts  of  a  right-angled  triangle  may  be  found 
by  either  of  the  four  last  cases :  or,  if  two  of  the  sides  are 
given,  by  means  of  the  property  that  the  square  of  the  hypo- 
thenuee  is  equivalent  to  the  sum  of  the  squares  of  the  two  othei 
sides.     Or  the  parts  may  be  found  by  Theorems  iv.  and  v. 

EXAMPLES. 

1.  In  a  right-angled    triangle  BAC, 
there  are  given    the   hypothenuse  BC 
-  250,  and  the  base  AC  =  240:  re-  C 
quired  the  other  parts. 


TRIGONOMETRY.  201 


Applications. 

To  find  the  angle  B. 

Afl          BC 

250          .    ar  comp. 

7.602060 

:             AC 

240          ... 

2.380211 

:  :     sin  ^ 

90°          ... 

10.000000 

sin  B 

73°  44'  23"     . 

9.982271 

Bttt  C  =  90°  —  i?  =  90°  —  73°  44'  23"  =  16°  15'  37"  : 

Or    C  may    be  found  from  the  proportion. 

Afl            CB  250              ar.  comp.         .  7.602060 

AC  240          ...         .  2.380211 

B                 10.000000 

008      C  16°  15'  37"     .         .         .  9.982271 


To  find  side  AB  by  Theorem  Iv. 

As       R  ar.  comp.  .  0.000000 

tan    C  16°  15'  37"     .         .         .  9.404889 

AC  240  ...         .  2.380211 

AB  70.0003  ....  1.845100 


2.  In   a  right-angled  triangle  BAC,  there  are  given  AC  ^ 
384,  and  B  =  53°  08' :  required  the  remaining  parts. 

Ans.  AB  =  287.96  ;  BC  =  479.979  ;   C  =  36°  62'. 

DEFINITIONS. 

1.  A  horizontal  angle  is  one  whose  sides  are  horizontal ;  its 
plane  is  also  horizontal. 

2.  An  angle  of  elevation  or  depression,  has  one  horizontal  sid*'. 
and  the  other  oblique,  but  lying  directly  above  or  below  the  first 


202 


TRIGONOMETRY 


Applications. 


APFLICATION     TO     HEIGHTS     AND     DISTANCKS. 


PROBLEM    I. 

To  determine  the  horizontal  distance   to  a  point  which  is  inar. 
cessible  by  reason  of  an  intervening  river. 

Let  ^  be  the  point.  Measure 
along  the  bank  of  the  river  ^  hori- 
zontal base  line  AB^  ana  select  the 
fttations  A  and  -5,  in  such  a  manner 
that  each  can  be  seen  from  the  other, 
and  the  point  C  from  both  of  them.  ^^S 
Then  measure  the  horizontal  angles  a 
CAB  and   CBA^  with  an  instrument  adapted  to  that  purpose. 

Let    us    suppose  that  we    have    found    AB  —  600  yarda, 
CAB  =  51°  35'  and   CBA  =  64°  51'. 


The  angle  C  =  180°  —  (A  +  B)  =  57°  34\ 
To  find  the  distance  BC, 


As 

sin  C 

51°  34'         ar.  comp. 

0.073649 

: 

sin  A 

57°  35'     ...         . 

9.926431 

•  • 

AB 

600            .... 

2.778151 

• 

BC 

600.11  yards. 
To  find  the  distance  AC, 

2.778231 

Ab 

sin  0 

57°  34'         ar.  comp. 

0.073649 

sin  B 

64°  51'              ... 

9.956744 

: : 

AB 

600            .                  ;         . 

2.778151 

AG 

643.94  yards.   . 

2.808544 

TBJGONOMETRY.  203 

ApplioatiouB. 

PROBLEM    II. 

To   determine   the   altitude  of  an    inaccessible   object   above    c 
given  horizontal  plane. 

FIRST    METHOD 

Suppose  2)  to  be  the  inaccessible  j) 

object,  and  BC  the  horizontal  plane  "'''"'' Js\ 

from    which   the   altitude   is    to    be  r» --'"i^        ^-*^^^^'/^ 
estimated:  then,  if  we  suppose  DC      V  /'  /  j j; 

to  be   a  vertical  line,  it  will  repre-  \  'vC'-'' 

sent  the  required   distance.  X^''"' 

Measure  any  horizontal  base  line,  as  BA\  and  at  the  ex- 
tremities B  and  A,  measure  the  horizontal  angles  CBA  and 
CAB.     Measure  also,  the  angle  of  elevation  DBC. 

Then  in  the  triangle  CBA  there  will  be  known,  two  angles 
and  the  side  AB\  the  side  BC  can  therefore  be  determined. 
Having  found  BC^  we  shall  have,  in  the  right-angled  triangle 
DBC^  the  base  BC  and  the  angle  at  the  base,  to  find  the  per- 
pendicular 7)  (7,  which  measures  the  altitude  of  the  point  D 
above  the  horizontal  plane  BC. 

Let  us  suppose  that  we  have  found 
BA  —  YSO  yards,  the  horizontal  angle  CBA  =41°  24', 
the  horizontal  angle  CAB  =  96°  28',  and  the  angle  of  eleva- 
tion J)BC^\0°  43'. 

In  the  triangle  BCA^  to  find  the  horizontal  distance  BC, 
The  angle  ^C^  =  ] 80°  —  (41°  24'  +  96°  28')  =  42°  08'=  ( 
As    sin  C         .     42°  08'         ar.  comp.         .         0.173369 
:        sin  ^         .     96°  28'     .         .         .         .         9.997228 
:  :  ^j5      .     780  ....         2.892095 

:  BC      .     1165.29    ....         8.062692 


^04  TRIGONOMETRY, 


Applioations. 


In  the  right-aDgled   triangle  DBO^  to  find  DC. 

As           R  ar.  comp.      .         ,  0.000000 

tan    DBC  10°  43'           ...  9.277043 

BG  1155.29         .         .         .  3.062692 

DC  218.64            .         .         .  2.339735 

Remark  I.  It  might,  at  first,  appear  that  the  solution  which 
we  have  given,  requires  that  the  points  B  and  A  should  be  in 
the  same  horizontal  plane ;  but  it  is  entirely  independent  of 
such  a  supposition. 

For,  'the  horizontal  distance,  which  is  represented  by  BA^ 
is  the  same,  whether  the  station  A  is  on  the  same  level  with 
B^  above  it,  or  below  it.  The  horizontal  angles  CAB  and 
CBA  are  also  the  same,  so  long  as  the  point  C  is  in  the  verti- 
cal line  D  C.  Therefore,  if  the  horizontal  line  through  A  should 
cut  the  vertical  line  D  C,  at  any  point  as  E^  above  or  below  C, 
AB  would  still  be  the  horizontal  distance  between  B  and  A, 
and  AE  which  is  equal  to  AC^  would  be  the  horizontal  dis- 
tance between  A  and   C. 

If  at  A,  we  measure  the  angle  of  elevation  of  tie  point  D 
wo  shall  know  in  the  right-angled  triangle  DAE,  the  base  AE, 
and  the  angle  at  the  base ;  from  which  the  perpendicular  D^ 
can  be  determined. 


TRIGONOMETRY.  208 

Application  B. 

Let  us  suppose  that  we  bad  measured  the  angle  of  elevation 
DAE,  and  found  it  equal  to  20°  15'. 

First:  In  the  triangle  BAC,  to  find  AC  or  its  equal  AE, 

As   sin   C  42°  08'     ar.  comp.         .         0.1733G9 

:     sin  B  41°  24'  ...         9.820406 

::        AB  780  ...         2.892095 

AC  768.9  .         .         .         2.885870 

In  the  right-angled  triangle  DAE,  to  find  DE. 

A3          R  ar.  comp.         .         .  0.000000 

tan  A  20°  15'           ...  9.5G6932 

AE  768.9              .         .         .  2.885870 

DE  283.66            .         .         .  2.452802 

Now,  since  DC  is  less  than  DE,  it  follows  that  the  station 
B  is  above  the  station  A,     That  is, 

DE  -  DC=  283.66  —  218.64  =  65.02  =  EC, 
which   expresses   the   vertical  distance    that    the   station  B  is 
above  the  station  A. 

Remark  II.  It  should  be  remeqjbered,  that  the  vertical  dia 
tance  which  is  obtained  by  the  calculation,  is  estimated  from 
a  horizontal  line  passing  through  the  eye  at  the  time  of  ob- 
Bcrvation.  Hence,  the  height  of  the  instrument  is  to  be  added, 
in  order  to  obtain  the  true  result. 

SECOND    METHOD. 

When  the  nature  of  the  ground  will  admit  of  it,  measure  a 
base  line  AB  in  the  direction  of  the  object  D.  Then  mea 
sure  with  the  instrument  the  angles  of  elevation  at  A  and  B. 

Then,  smce  the  outward  angle  DBC  is  equal  to  the  sum 
18 


206  TRIGONOMETRY. 

Applications. 

of  the  angles  A  and 
ABB,  it  follows,  that 
the  angle  ADB  is 
equal  to  the  diflference 


of  the    angles  of  ele-    /^ — ^c 

vation  at  A  and  B»  Hence,  we  can  find  all  the  parts  of  the 
triangle  ADB.  Having  found  DB,  and  knowing  the  angle 
DBC,  we  can  find  the  altitude  DC, 

This  method  supposes  that  the  stations  A  and  B  are  on 
the  same  horizontal  plane ;  and  therefore  can  only  be  used 
when  the  line   AB  is   nearly  horizontal. 

Let  us  suppose  that  we  have  measured  the  base  line,  and 
the  two  angles  of  elevation,  and 

!AB  =  9T5  yards, 
A  =  15°  36', 
DBC=  27°  29'; 
required  the  altitude  DC. 


First:  ADB  =  DBC -  A  =  27°  29'  -  15°  36'  =  11°  58' 

In  the  triangle  ADB,  to  find  BD. 

As   sin  D  11°  53'      ar.  comp.         .  0.686302 

sin^  15°  36'  .         .  .  9.429623 

AB  975        ...         .  2.989005 

DB  1273.3  .         .         .  3.104930 

In  the  triangle  DBC,  to  find  DC. 

AS  B  ar.  comp.         .  0.000000 

sin  B  27°  29'  ...  9.664163 

DB  1273.3  .         .         .  3.104930 

DC  587.61  .         .         .  2.76909a 


TRIGONOMETRY. 


207 


Applicationn. 


PROBLEM    III. 

To  determine  the  perpendicular   distance   of  an  object  below  a 
given  horizontal  plane. 

Suppose    C  to  be   directly    over      ,  ^^^ 

tlie  given  object,  and  A  the  point     P^^^^  yX       \ 
through  which  the  horizontal  plane 
is  supposed  to  pass. 

Measure    a   horizontal   base   line  ^^/^ 

AB,  and  at  the  stations  A  and  B  ^^ 
conceive    the   two   horizontal    liues      '""^ 

AC,  BC^  to  be  drawn.  The  oblique 
lines  from  A  and  B  to  the  object  will  be  the  hypothenusca 
of  two  right-angled  triangles,  of  which  AC,  BC,  are  the 
bases.  The  perpendiculars  of  these  triangles  will  be  the  dis- 
tances from  the  horizontal  lines  AC,  BC,  to  the  object.  If 
we  turn  the  triangles  about  their  bases  AC,  BC,  until  the}' 
become  horizontal,  the  object,  in  the  first  case,  will  fall  at  C\ 
and  in  the  second  at  C". 

Measure   the  horizontal   angles    CAB,    CBA,   and  also    tho 
angles  of  depression   C'AC,  C'BC 

Let  us  suppose  that  we  have 

AB  =  672  yards 

BAC  --=  Y2°  29' 

found  I   ABC  =  39°  20' 

C'AC  =2r  49' 

,  C'BC  =  19°  10' 

First:   In  the  triangle  ABC,  the  horizontal  angle  ACB  =a 
180«  -  (AA-  B)  =  180°  -  111°  49'  =  68°  11'. 


208  TR3GONOMETR7 

Applications. 


Ab  sin  C 

68°  11' 

.      sin  A 

72°  29' 

j:         ab 

672 

BO 

690.28 

To  find  the  horizontal  distance  AC. 

As   sin   0  68°  11'      ar.  comp.         ,  0.032276 

:       Bin  B  39°  20'  ...  9.801973 

J  :  AB  672  ...  2.827369 

AO  458.79  .         .         .  2.661617 

To  find  the  horizontal  distance  BC, 

ar.  comp.     ,         .  0.032275 

.  9.979380 

.  2.8273C9 

.  2.839024 

In  the  triangle   CAC,  to  find   C(7', 

Ab         B  .         ar.  comp.         .         .  0.000000 

tan  C'AC       27°  49'     .         .         .         .  9.722315 

AO  458.79      .         .         ;         .  2.661617 

CC  242.06      ....  2.383932 

In  the  triangle   C7^(7",  to  find   00" 

As         i?  .         ar.  comp.         .        .  0.000000 

;      tan  0"B0       19°  10'    .         .         .  9.541061 

.:  BO  690.28      ....  2.839024 

:  00"  239.93  .         .  2.380083 


TRIGONOMETRY 


209 


Applications. 


Hence  also,  CC  -  CC"  =  242.06  -  239.93  =  2.13  yards, 
which  is  the  height  of  the  station  A  above  station  £. 

PROBLEMS. 

1.  Wanting  to  know  the  distance  between  two  inaccessible 
objects,  which  lie  in  a  direct  line  from  the  bottom  of  a  tower 
of  120  feet  in  height,  the  angles  of  depression  are  measured, 
and  are  found  to  be,  of  the  nearer  57°,  of  the  more  remote 
26°  30' :  required  the  distance  between  them. 

Ans.  173.656  feet. 

2.  In  order  to  find  the  distance  between 
two  trees  A  and  B^  which  could  not  be 
directly  measured  because  of  a  pool  which 
occupied  the  intermediate  space,  the  dis- 
tances of  a  third  point  C  from  each  of 
them  were  measured,  and  also  the  included 
angle  ACBi  it  was  found  that 

CB  =  672  yards 
(7i4  =  688  yards 
ACB  =  55°  40'; 
required  the  distance  AB. 

Ans.  592.967  yards. 

3.  Being  on  a  horizontal  plane,  and  wanting  to  ascertain 
the  height  of  a  tower,  standing  on  the  top  of  an  inaccessible 
hill,  there  were  measured,  the  angle  of  elevation  of  the  top 
of  the  hill  40°,  and  of  the  top  of  the  tower  51";  then  mea- 
suring in  a  direct  line  180  feet  farther  from  the  hill,  the  angle 
of  elevation  of  the  top  of  the  tower  was  33°  45' ;  required  tLe 
height  of  the  towei. 

Ans.  83.998  feet. 
18* 


210 


TRIGONOMETRY. 


Applications. 


4.  Wanting  to  know  the  horizon- 
tal distance   between    two    inaccessi-  ^-Q*^ 


ble  objects   E  and  TT,  the  following 
caeasnreraents  were  made, 

f  AB  —  636  yards 
BAW  =  40°  16' 
Mz.  ^    WAE  =  5V°  40' 
ABE  =  42°  22' 
EBW=  71°  07' 
required  the  distance  EW. 


A71S.  939.634  yards. 


5.  Wanting  to  know  the 
liorizontal  distance  between  two 
inaccessible  objects  A  and  B,  ' 
and  not  finding  any  station 
from  which  both  of  them  could 
he  seen,  two  points  C  and  D, 
were  chosen,  at  a  distance  from 
each  other,  equal  to  200  yards  ;  from  the  former  of  these  points 
A  could  be  seen,  and  from  the  latter  B,  and  at  each  of  the 
points  C  and  D  a  staff  was  set  up.  From  C  a  distance  CF 
was  measured,  not  in  the  direction  DC,  equal  to  200  yards, 
and  from  J)  a  distance  DE  equal  to  200  yards,  and  the  follow- 
ing anglef  taken, 

{AFC  =  83°  00'   BBE  =  54°  30' 
ACD  =  63°  30'   BDC  =  156°  25' 
ACF=  64°  31'   BED  =  88°  30' 

Ans.  AB  =  345.467  yards. 


APPLICATIONS 

OF 

GEOMETRY. 

MENSURATION      OF      SURFACES. 
DEFINITIONS. 

I  The  area  of  any  figure  has  already  been  defined  to  be 
the  measure  of  its  surface  (Bk.  IV.  Def.  7).  This  measure  is 
merely  the  number  of  squares  which  the  figure  contains. 

A  square  whose  side  is  one  inch,  one  foot,  or  one  yard, 
(fee,  is  called  the  measuring  unit ;  and  the  area  or  contents  of 
a  figure  is  expressed  by  the  number  of  such  squares  which 
the  figure  contains. 

2.  In  the  questions  involving  decimals,  the  decimals  are 
generally  carried  to  four  places,  and  then  taken  to  the  nearest 
figure.  That  is,  if  the  fifth  decimal  figure  is  5,  or  greater 
than  5,  the  fourth  figure  is  increased  by  one. 

3.  Sur\'eyors,  in  measuring  land,  generally  use  a  chain 
called  Gunter's  chain.  This  chain  is  four  rods,  or  66  feet  in 
length,  and  is  divided  into  100  links. 

4.  An  acre  is  a  surface  equal  in  extent  to  10  square  chains; 
that  is,  equal  to  a  rectangle  of  which  one  side  is  ten  chaino 
and  the  otlier  side  one  chain. 

One  quarter  of  an  acre,  is  called  a  "ood. 
Since  the  chain  is  4  rods  in  length,  1  square  cha'n  contains 
6  square  rods  ;  and  therefore,  an  acre,  which  is   10  square 
chains,  contains   160  square  rods,  and   a  rood   contains   40 
square  rods.     Tlie  square  rods  are  called  perches. 


212  APPLICATIONS 

Mensuration    of    Surfaces. 

5.  Land  is  generally  computed  in  acres,  roods,  and  perchoa 
which  are  respectively  designated  by  the  letters  A,  R,  P. 

When  the  linear  dimensions  of  a  survey  are  chains  oi  liake 
the  area  will  be  expressed  in  square  chains  or  square  links, 
and  it  is  necessary  to  form  a  rule  for  reducing  this  area  tc 
acres,  roods,  and  perches.  For  this  purpose,  let  }:s  form  llie 
following 

TABLE. 

1  square  chain  =:  1 00  x  1 00  =  1 0000  square  links. 
I  acre  =  10  square  chains  =  100000  square  links 

1  acrer=4  roods=zl60  perches. 
1  square  mile  =  6400  square  chains  =  640  acres. 

6.  Now,  when  the  linear  dimensions  are  links,  the  area 
will  be  expressed  in  square  links,  and  may  be  reduced  to 
acres  by  dividing  by  100000,  the  number  of  square  links  in  an 
acre :  that  is,  by  pointing  off  five  decimal  places  from  th*' 
right  hand. 

If  the  decimal  part  be  then  multiplied  by  4,  and  five  places 
of  decimals  pointed  ofi*  from  the  right  hand,  the  figures  to  the 
left  hand  will  express  the  roods. 

If  the  decimal  part  of  this  result  be  now  multiplied  by  40, 
and  five  places  for  decimals  pointed  off,  as  before,  the  figures 
to  the  left  will  express  the  perches. 

If  one  of  the  dimensions  be  in  links,  and  the  other  in  chains, 
(he  chains  may  be  reduced  to  links  by  annexing  two  ciphers, 
or,  the  multiplication  may  be  made  without  annexing  the  ci- 
phers, and  the  product  reduced  to  acres  and  decimals  of  an 
acre,  by  pointing  off  three  decimal  places  at  the  right  hand. 

Wlien  both  dimensions  are  in  chains,  the  ])rodnct  is  re- 


OF    GE  O  M  E  TR  Y.  213 

Mensuration    of    Surfaces. 

luccd  to  acres  by  dividing  by  1 0,  or  pointing  off  one  dcciraaJ 
place. 

From  which  we  conclude  :  that, 

I.  [f  links  be  mulliplied  by  links,  the  product  is  reduced  to 
ayres  by  pointing  off  Jive  decimal  places  from  the  right  hand. 

II.  If  chains  be  multiplied  by  links,  the  product  is  reduced  to 
acres  by  pointing  off  three  decimal  places  from  the  right  hand. 

III.  If  chains  be  multiplied  by  chains,  the  product  is  reduced 
to  acres  by  pointing  off  one  decimal  place  from  the  right  hand. 

7.  Since  there  are  16.5  feet  in  a  rod,  a  square  rod  is  equal 
to  1 6.5  X  1 6.5 =272.25  square  feet. 

If  the  last  number  be  multiplied  by  160,  we  shall  have 
272.25x160=43560  the  square  feet  in  an  acre. 

Since  there  are  9  square  feet  in  a  square  yard,  if  the  last 
number  be  divided  by  9,  we  obtain 

4840  =  the  number  of  square  yards  in  an  acre 

PROBLEM     I. 

To  find  the  area  of  a  square,  a  rectangle,  a  rhombus,  or  a 
paralleh)gram. 

RULE. 

Multiply  the  base  by  the  perpendicular  height  and  the  produd 
will  be  the  area  (Bk.  IV.  Th.  viii). 

EXAMPLES. 

I.  Required  the  area  of  the  square 
A  BCD,  each  of  whose  sides  is  36  feet 


214  APPLICATIONS 


Mensuration    of    Surfaces. 


We  multiply  two  sides  of 
the  square  together,  and  the 
product  is  the  area  in  square 

feet. 


Operation. 
36x36  =  1290   sq.  ft. 


2.  How  many  acres,  roods,  and  perches,  in  a  square  whose 
side  is  35.25  chains?  Ans.  124  A.  \  R.  \  P 

3.  Wliat  is  the  area  of  a  square  whose  side  is  8  feet  4 
inches  ?  Ans.  69  ft.  5'  4" 

4.  What  is  the  contents  of  a  square  field  whose  side  is  46 
rods?  Ans.  12  A.  0  R.  ^6  P. 

5.  Whit  is  the  area  of  a  square  whose  side  is  4769  yarde  ! 

Ans.  22743361  sq.  yds 


6.  What  is  the  area  of  the  parallelo- 
gram ABCD,  of  which  the  base  AB  is 
64  feet,  and  altitude  DE,  36  feet  ? 


D 


A~^ 


We  multiply  the  base  64, 
by  the  perpendicular  height 
36,  and  the  product  is  the  re- 
quired area. 


Operation. 
6ix36=2304  sq.ft. 


7.  What  is  the  area  of  a  parallelogram  whose  base  is  If^.gfi 
yards,  and  altitude  8.5  ?  Ans  104,125  sq.  yds. 

8.  What  is  the  area  of  a  parallelogram  whose  base  is  8.75 
chains,  and  altitude  6  chains  ?  Ans.  b  A.  \  R    OP. 

9.  What  is  the  area  of  a  parallelogram  whose  base  is  7  ''oot 

9  inclies,  and  altitude  3  feet  6  inches  ? 

Ans.  27  sq.ft.  1'  6' 


OF     GEOMETRY 


216 


Mensuration    of    Surfaceo 


10.  To  find  the  area  of  a  rectangle 
A  BCD,  of  which  the  base  AB=45 
yards,  and  the  altitude  AD=:\5  yards. 

Here  we  simply  multiply 
the  base  by  the  altitude,  and 
the  product  is  the  area. 


B 


Operation 
45xl5rzG75  sq.  yds. 


11.  What  is  the  area  of  a  rectangle  whose  base  is  14  feel 
6  inches,  and  breadth  4  feet  9  inches  ? 

Ans.  G8  sq.ft.  10'  6". 

12.  Find  the  area  of  a  rectangular  board  whose  length  is 
112  feet,  and  breadth  9  inches.  Ans.  84  sq.  ft. 

13.  Required  the  area  of  a  rhombus  whose  base  is    10.51 
and  breadth  4.28  chains.  Ans.  4  A.  \  R.  39.7  P+. 

14.  Required  the  area  of  a  rectangle  whose  base  is  12  feot 
6  inches,  and  altitude  9  feet  3  inches. 

Ans.  115  sq.  ft.  T  6" 

PROBLEM     II. 

To  find  the  area  of  a  triangle,  wheii  the  base  and  altitude 
are  known. 

RULE. 

I.  Multiply  the  base  by  the  altitude^  and  half  the  product  mil 
be  the  area. 

II.  Multiply  the  base  by  half  the  altitude  and  the  product  utill 
he  the  area  (Bk.  IV.  Th.  ix). 

EXAMPLES. 

I  Required  the  area  of  the  triangle 
ABC,  whose  base  yl5  is  10,75  foet, 
anJ  altitude  7,25  feel. 

15 


216  APPLICATIONS 


Mensuration    of    Surfaces. 


We  first  multiply  the  base 
by  the  altitude,  and  then  di- 
♦  ide  the  product  by  2. 


Operation. 
10,75x7,25  =  77,9375 

and 
77,9375-^2  =  38,96875 
=area 
2    What  is  the  area  of  a  triangle  whose  base  is  18  feet  4 
inches,  and  altitude  11  feet  10  inches  ? 

Ans    108  sq.  ft.  5'  8". 

3.  What  is  the  area  of  a  triangle  whose  base   is   12.25 
chains,  and  aUitude  8.5  chains?  Ans.  b  A.  OR.  33  P. 


4.  What  is  the  area  of  a  triangle  whose  base  is  20  feet, 
and  altitude  10.25  feet.  Ans.  102.5  sq.  ft. 

5.  Find  the  area  of  a  triangle  whose  base  is  625  and  alti- 
tude 520  feet.  Ans.  162500  sq.  ft 

6.  Find  the  number  of  square  yards  in  a  triangle  whose 
base  is  40  and  ahitude  30  feet.  Ans.  66^  sq.  yds. 

7.  What  is  the  area  of  a  triangle  whose  base  is  72.7  yards, 
and  altitude  36.5  yards?  Ans.  1326,775  sq.  yds 

PROBLEM    III. 

To  find  the  area  of  a  triangle  when  the  three  sides  are 
known. 

RULE, 

}.  Add  the  three  sides  together  and  take  half  their  sum, 

I I .  From  this  half  sum  take  each  side  separately. 

III.  Multiply  together  the  half  sum  and  each  of  the  three 
remainders,  and  then  extract  the  square  root  of  the  product^ 
which  will  he  the  required  area. 


OF     G  E  O  M  E  r  R  Y.  217 


Mensuration    of    Surfacea. 

EXAMPLES. 

1.  Find  the 

area  of  a  triangle  whose  sides  are  20,  30,  and 

10  rods. 

20 
30 

45                          45                          45 

20                          30                          40 

40 
2)<)0 
45  lialf  sum 

25  1^^  rem.            15  2d  rem.              5  3d  rem 

Then,  to  obtain  the  product,  we  have 

45x25x15x5  =  84375; 
from  which  we  find 


area=  -/84375 =290,4737  perches. 

2.  How  many  square  yards  of  plastering  are  there  in  a  tri- 
angle, whose  sides  are  30,  40,  and  50  feet?  Ans.  66j. 

3.  The   sides  of  a  triangular  field  are  49   chains,  50.25 
chains,  and  25.69  :  what  is  its  area  ? 

Ans.  61  A.  1  R.  39,68  P 

4.  What  is  the  area  of  an  isosceles  triangle,  whose  base  ia 
20,  and  each  of  the  equal  sides  15  ?  Ans.  Ill  803. 

5.  How  many  acres  are  there  in  a  triangle  whose  three 
sides  are  380,  420  and  765  yards.      Ans.  0  A.  OR.  38  P. 

6.  How  many  square  yards  in  a  triangle  whose  sides  are 
13,  14,  and  15  feet.  Ans.  91. 

7    What  is  the  area  of  an  equilateral  triangle  whose  side 
is  25  feet  ?  Ans.  270.6329  sq.  ft. 

8.  What  is  the  area  of  a  triangle  whoso  sides  are  24,  3G, 
and  48  yards?  Ans   418.282  sq.  yds. 


218  APPLICATIONS 

Mensuration    of    Surfaces. 

PROBLEM    IV. 

To  find  the  hypothenuse  of  a  right  angled  triangle  when 
the  base  and  perpendicular  are  known 

RULE. 

I..  Square  each  of  the  sides  separately. 

IT.  Add  the  squares  together. 

III.  Extract  the  square  root  of  the  sum,  which  will  be  the  hy- 
'pothcnuse  of  the  triangle  (Bk.  IV.  Th.  xii). 

EXAMPLES. 

1.  In  the  right  angled  triangle  ABC, 
we  have,  ^5  =  30  feet,  BC—AQ  feet,  to 
find^lC. 


We  first  square  each  side, 
and   then   take   the    sum,  of 
which  we  extract  the   square 
root,  which  gives 

A 

Operation. 

30^=   900 
40^=:  1600 

^C=-v/2500  =  50  feet. 

2.  The  wall  of  a  building,  on  the  brink  of  a  river,  is  120 
feet  high,  and  the  breadth  of  the  river  70  yards  :  what  is  the 
length  of  a  line  which  would  reach  from  the  top  of  the  wall  to 
the  opposite  edge  of  the  river?  Ans.  241.86  ft. 

3.  The  side  roofs  of  a  house  of  which  the  caves  are  of  the 
same  height,  form  a  right  angle  at  the  top.  Now,  the  length 
of  the  rafters  on  one  side  is  10  feet,  and  on  the  other  14  feet : 
what  is  the  breadth  of  the  house  ?  Ans.  1 7.204  ft. 

4.  WhU  would  be  the  width  of  the  house,  in  the  last  ex- 
ample,  if  the  rafters  on  each  side  were  10  feet? 

Ans.  14.142  ft. 


OF      GEOMETRY. 


219 


Mensuration    of    Surfaces. 


5.  What  would  be  the  width,  if  the  rafters   on  each  side 
were  14  feet  ^  Ans.  19.7989  ft. 

PROBLEM    V. 

When  the  hypothenuse  and  one  side  o(  a  right  angled  tri- 
angle are  knou  n,  to  find  the  other  side 


Square  the  hypothenuse  and  also  the  other  given  side,  and 
take  their  difference :  extract  the  square  root  of  this  difference^ 
and  the  result  will  be  the  required  side  (Bk.  IV.  Th.  xii.  Cor.). 

EXAMPLES. 

1.     In  the  right  angled  triangle -4  jBC, 
there  are  given 

AC  =  50  feel,  and  ^5  =  40  feet, 
required  the  side  BC. 

We  first  square  the  hypoth- 
enuse and  the  other  side,  after 
wliich  we  take  the  difierence, 
and  then  extract  the  square 
root,  which  gives 

Z?C=-/900=30  feet. 

2  The  height  of  a  precipice  on  the  brink  of  a  river  is  103 
feet,  and  a  line  of  320  feet  in  length  will  just  reach  from  the 
top  of  it  to  the  opposite  bank :  required  the  breadth  of  the 
river.  Ans.  302.9703  ft. 

3.  The  hypothenuse  of  a  triangle  is  53  yards,  and  the  per 
pendiciilar  45  yards  :  what  is  the  base  ?  Ans.  28  yds. 

4     \  ladder  60  feel  in  len^h,  will  reach  to  a  windoAv  40 


Difl'-^r   900 


2'20  A  P  P  L  I  C  A  T  I  O  IN  S 


Mensuration    of    Surfaces 


feet  from  the  ground  on  one  side  of  the  street,  and  by  tnniiug 
it  over  to  the  other  side,  it  will  reach  a  window  50  feet  from 
the  ground :  required  the  breadth  of  the  street. 

Ans.  77.8875  fi. 

PROBLEM    VI. 

To  find  the  area  of  a  trapezoid. 

RULE. 

Multiply  the  sum  of  the  parallel  sides  hy  the  perpendicular 
distance  between  them,  and  then  divide  the  product  by  two :  the 
quotient  will  be  the  area  (Bk.  IV.  Th.  x). 

EXAMPLES. 


1 .  Required  the  area  of  the  trapezoid 
ABCD,  having  given 


AJ5=321.51  feet,  DC=2U.24  f^et,  and  CE=zl7lA6  feei 

Operation. 


We  first  find  the  sum  of  the 
sides,  and  then  multiply  it  by 
the  perpendicular  height,  after 
which,  we  divide  the  product 
by  2,  for  the  area. 


321.514-214.24=535.75- 
sum  of  parallel  sides. 

Then, 
535.75x171.16  =  91698.97 

and,   ?i^?!:?Z  =  45849.485 
2 

I  =the  area. 


2  What  is  the  area  of  a  trapezoid,  the  parallel  sides  of 
which,  are  12.41  and  8.22  chains  and  the  perpendicular  dis- 
tance between  them  5.15  chains  ? 

Ans.  5  A.l  R.  9.956  P. 

3  Required  the  area  of  a  trapezoid  whose  parallel  sides 


O  F     G  E  O  M  E  T  R  Y  .  221 

Mensuration    of    Surfaces. 

are  25  feet  6  inches,  and  18  feet  9  inches,  and  the  perpen- 
dicular distance  between  them  10  feet  and  5  inches. 

Ans.  230  sq.  ft.  5'  7". 

4.  Required  the  area  of  a  trapezoid  whose  parallel  sides 
are  20.5  and  12.25,  and  the  perpendicular  distance  between 
them  10.75  yards.  Ans.  176.03125  sq.  yds. 

5.  What  is  the  area  of  a  trapezoid  whose  parallel  sides  are 
7.50  chains,  and  12.25  chains,  and  the  perpendicular  height 
15.40  chains  ?  Ans.  15  A.  0  R.  33.2  P 

PROBLEM     VII. 

To  find  the  area  of  a  quadrilateral. 

RULE. 

Measure  the  four  sides  of  the  quadrilateral,  and  also  one  of  the 
diagonals :  the  quadrilateral  will  thus  be  divided  into  two  trian* 
gles,  in  both  of  which  all  the  sides  will  be  known.  Then,  find 
the  areas  of  the  triangles  separately,  and  their  sum  will  be  tht 
area  of  the  quadrilateral. 

EXAMPLES. 

1.  Suppose  that  we  have  meas- 
ured the  sides  and  diagonal  A  C,  of 
the  quadrilateral  ABCD,  and  found 

AT 
^5=40.05  chains;   CD =29.87  chains, 
BC =26.27  chains    ^D  =  37.07  chains, 
and  ^C= 55  chains: 

required  the  area  of  the  quadrilateral 

^^^  Ans.  10\   A.  1   R    16  P 


Ii22 


APPLICATIONS 


Mensi. ration    of    Surfaces. 

Remark. — Instead  of  measuring 
tlie  four  sides  of  the  quadrilateral, 
we  may  let  fall  the  perpendicu- 
lirs  Bbj  Dgj  on  the  diagonal  AC. 
The  area  of  the  triangles  may  then 
be  determined  by  measuring  these 
perpendiculars  and  diagonal  AC.  The  pendiculars  are^Dg  — 
18.95  chains,  and  Bb  =17.92  chains. 

2.  Required  the  area  of  a  quadrilateral  whose  diagonal  is 
B0.5,  and  two  perpendiculars  24.5,  and  30.1  feet. 

Ans.  2197.65  sq.ft. 

3.  What  is  the  area  of  a  quadrilateral  whose  diagonal  is 
108  feet  6  inches,  and  the  perpendiculars  56  feet  3  inches, 
and  60  feet  9  inches  ?  Ans.  6347 sq.ft.  3  . 

4.  How  many  square  yards  of  paving  in  a  quadrilateral 
whose  diagonal  is  65  feet,  and  the  two  perpendiculars  28,  and 
33i  feet  ?  Ans.  222^2  sq.  yds. 

5.  Required  the  area  of  a  quadrilateral  whose  diagonal  is 
42  feet,  and  the  two  perpendiculars  18,  and  16  feet. 

Ans.  714  sq.  ft. 

6.  What  is  the  area  of  a  quadrilateral  in  which  the  diago- 
nal is  320.75  chains,  and  the  two  perpendiculars  69.73  chains, 
and  130.27  chains  ?  Ans.  3207  A.  2  R. 


PROBLEM     VIII. 

To  find  the  area  of  a  regular  polygon. 

RULE. 


Multiply  half  the  perimeter  of  the  figure  by  the  perpendicular 
Let  fall  from  the  centre  on  one  of  the  sides,  and  the  product  icill 
he  the  area,  (Bk.  IV.  Th.  xxvi) 


OP     GEOMETRY 


223 


Mensuration    of    Surfaces 


EXAMPLES. 


1.  Required  the  area  of  the  regular 
pentagon  ABiWE,  each  of  whose 
sides  AB,  EC,  &c.,  is  25  feet,  and 
the  perpendicuhir  OP,  17.2  feet 


We  first  multiply  one  side 
by  the  number  of  sides  and 
divide  the  product  by  2  :  this 
gives  half  the  perimeter  which 
we  multiply  by  the  perpen- 
dicular for  the  area. 


Operation. 

?5^=62.5=half  the  penny. 

eter.     Then, 

62.5x17.2  =  1075  ^^. /^=the 

area. 


2.  The  side  of  a  regular  pentagon  is  20  yards,  and  the  per- 
pendicular from  the  centre  on  one  of  the  sides  13,76382  ;  re- 
quired the  area. 

Ans.  688.191   sq.  yds. 

3.  The  side  of  a  regular  hexagon  is  14,  and  the  perpen- 

licular  from  the  contre  on  one  of  the  sides  12.1243556:  re- 

]uired  the  area. 

Ans.  509.2229352  sq.ft. 

4.  Required  the  area  of  a  regular  hexagon  whose  side  is 
14.6,  and  perpendicular  from  the  centre  12.64  feet. 

Ans.  553.632  sq   ft. 

5.  Required  the  area  of  a  heptagon  whoae  side  is   19,38 

arid  perpendicular  20  feet. 

Ans.  1 356.6  sq.  ft. 

The  following  table  shows  the  areas  of  the  ten  regular 


224 


APPLICATIONS 


Mensuration    of    Surfaces. 


polygons  when  the  side  of  each  is  equal  to  1 :  it  also  shows 
the  length  of  the  radius  of  the  inscribed   circle. 


Number  of 

sides. 

Names. 

Areas. 

Radius  of  inscnbcd 
circle. 

3 

Triangle, 

0.4330127 

0.2886751 

4 

Square, 

1.0000000 

0.5000000 

5 

Pentagon, 

1.7204774 

0.6881910 

6 

Hexagon, 

2.5980762 

0.8660254 

7 

Heptagon, 

3.6339124 

1.0382617 

8 

Octagon, 

4.8284271 

1.2071068 

9 

Nonagon, 

6.1818242 

1.3737387 

10 

Decagon, 

7.6942088 

1.5388418 

11 

Undecagon, 

9.3656404 

1.2028437 

12 

Dodecagon, 

11.1961524 

1.8660254 

Now,  since  the  areas  of  similar  polygons  are  to  each  othei 
as  the  squares  described  on  their  homologous  sides  (Bk.  IV 
Th.    xx),  we  have 

1       :     tabular  area     :  :     any  side  squared     :     area. 

Hence,  to  find  the  area  of  a  regular  polygon,  we  have  the 
following 


I     Square  the  side  of  the  polygon. 

n.  Multiply  the  square  so  founds  by  the  tahular  area  set  oppo- 
site the  polygc^  of  the  same  number  of  sides,  and  the  product 
will  be  the  irea. 

EXAMPLES. 

1 .  What  is  the  area  of  a  regular  hexagon  whose  side  is  20 
20^ = 400         and  tabular  area = 2 ,5980762. 

Hence, 

2.5980762  X  400= 1039.23048=the  area. 


OF     GEOMETRY 


225 


Mensuration    of    Surfaces. 


2.  What  is  the  area  of  a  pentagon  whose  side  is  25  ? 

Arts.  1075.298375. 

3.  What  is  the  area  of  a  heptagon  whose  side  is  30  feet 

Ans.  3270.52116 

4.  What  is  the  area  of  an  octagon  whose  side  is  10  feet  \ 

Ans.  482.84271   sq.  ft 
b.  The  side  of  a  nonagon  is  50  :  what  is  its  area  ? 

Ans.   15454.5605 

6.  The  side  of  an  undecagon  is  20  :  what  is  its  area  ? 

Ans.  3746.25616. 

7.  The  side  of  a  dodecagon  is  40  :  what  is  its  area  ? 

Ans.  17913.84384 


PROBLEM    IX. 


To  find  the  area  of  a  long  and  irregular  figure,  boiuided  on 
one  side  by  a  straight  line. 

RULE. 

I.  Divide  the  right  line  or  base  into  any  number  of  equal 
parts,  and  measure  the  breadth  of  the  figure  at  the  points  of  di 
vision,  and  also  at  the  extremities  of  the  base. 

II.  Add  together  tlie  intermediate  breadths^  and  half  the  svm 
of  the  extreme  ones 

III.  Multiply  this  sum  by  the  base  line,  and  divide  the  produd 
bif  th^  number  of  equal  parts  of  the  base. 

EXAMPLES. 

1 .  The  breadths  of  an  irregu- 
lar   figure,   at    five    equidistant       % 
places,  A,  B,  C,  P,  and  E,  be- 
ing  8.20    chains.    7.40    chainn. 


2'2(i  APPLICATIONS 

Mensuration    of    Surfaces. 

9.20  chains,   10.20  chains,  and  8.60  chains,  and  the  whole 
length  40  chains  :  required  the  area. 

8.20  35.20 

8.60  40 

2)lK8Q  4)1408.00 

8,40  mean  of  the  extremes.    352.00  square   chains. 

7.40 

9.20 
10.20 
35.20  thd  sum. 

Ans.  35  A.  32  P. 

2.  The  length  of  an  irregular  piece  of  land  being  21  chains 
and  the  breadths,  at  six  equidistant  points,  being  4.35  chains 
5.15  chains,  3.55  chains,  4.12  chains,  5.02  chains,  and  6.10 
chains :  required  the  area.  Ans.  9  A.  2  R.  30  P. 

3.  The  length  of  an  irregular  figure  is  84  yards,  and  the 
breaoths  at  six  equidistant  places  are  17.4 ;  20.6  ;  14.2  ;  16.5; 
20.1  ;   and  24.4  :   what  is  the  area  ?     Ans.  1550.64  sq.  yds. 

4.  The  length  of  an  irregular  field  is  39  rods,  and  its 
breadths  at  five  equidistant  places,  are  4.8  ;  5.2  ;  4.1  ;  7.3 , 
and  7.2  rods  :  what  is  its  area  ?  Ans.  220.35  sq.  rods. 

5.  The  length  of  an  irregular  field  is  50  yards,  and  its 
breadths  at  seven  equidistant  points,  are  5.5 ;  6.2  ;  7.3  ;  6 ; 
7.5  ;  7  ;  and  8.8  yards  :  what  is  its  area  ? 

Ans.  342.916  sq.  yds. 

6.  The  length  of  an  irregular  figure  being  37.6,  and  the 
breadths  at  nine  equidistant  places,  0;  4.4  ;  6.5  ;  7,6  ;  5.4  ;  8; 
5.2  ;  6.5  ;  and  6 J  :  what  is  the  area?  Ans.  219.255. 

PROBLEM     X. 

To  find  the  circumference  of  a  circle  when  the  diameter  is 
known. 


OFGEOMETRY.  221 

Mensuration   of    Surfaces. 

RULE 

Multiply  the  diameter  by  3.1416,  and  the  product  will  te  tht 
circumference. 

EXAMPLES. 

1.   What  is  the  circumference  of  a  circle  whoso  diamelei 
is  17? 

Wc    simply   muhiply   the 


immbcr  3.1 41 G  by  the  diam- 
eter anl  the  product  is  the 
circumference 


Operation. 
3.1416x17  =  53.4072, 
which  is  the  circumference. 


2.  What  is  the  circumference  of  a  circle  whose  diameter  \& 
40  feet?  Ans.  125.664/11. 

3.  What  is  the  circumference  of  a  circle  whose  diameter  is 
12  feet  ?  Ans.  37.6992  ft. 

4.  What  is  the  circumference  of  a  circle  whose  diameter  is 
22  yards?  Ans.  69.1152  yds. 

5.  What  is  the  circumference  of  the  earth — the  mean  diam- 
eter being  about  7921  miles?  Ans.  24884.6136  mi. 

PROBLEM     XI. 

To  find  the  diameter  of  a  circle  when  the  circumference  is 
knowa 

RULE. 

Dimde   the  circumference  by  the  number   3.1416   and  the  qxxo^ 
tient  tmll  be  the  diameter. 

EXAMPLES. 

1.  The  circumference  of  a  circle  is  69.1152  yards:  whai 
is  the  diameter  * 


228  APPLICATIONS 


Mensuration    of    Surfaces. 


We  simply  divide  the  cir- 
cumference by  3.1416,  and 
the  quotient  22  is  the  diam- 
eter sought. 


Operation. 

3.1416)691152(22 
62832 


62832 
62832 


2.  What  is  the  diameter  of  a  circle  whose  circumference  la 
11652.1944  feet  ?  Ans.  3709. 

3.  What  is  the  diameter  of  a  circle  whose  circumference  is 
6850?  Ans.  2180.4176. 

4.  What  is  the  diameter  of  a  circle  whose  circumference  is 
50?  ^n;?.  15.915. 

5.  If  the  circumference  of  a  circle  is  25000.8528,  what  is 
tlie  diameter  ?  Ans.  7958. 

PROBLEM     XII. 

To  find  the  length  of  a  circular  arc,  when  the  number  ol 
degrees  which  it  contains,  and  the  radius  of  the  circle  are 
known. 

RULE. 

Multiply  the  number  of  degrees  by  the  decimal  .01745,  and 
the  product  arising  by  the  radius  of  the  circle. 

EXAMPLES. 

1.  What  is  the  length  of  an  arc  of  30  degrees,  in  a  circle 
whose  radius  is  9  feet. 


We  merely  multiply  the 
given  decimal  by  the  number 
of  degrees,  anl  by  the  radius. 


Operation. 

,01745x30x9  =  4.7115, 

which  is  the  length  of  the  arc 


Remark. — When  the  arc  contains  degrees  and  minutes,  re- 
duce the  minutes  to  the  decimals  of  a  degree,  which  is  done 
by  dividing  thera  by  60. 


OF     GEOMETRY 


229 


Mensuration    of    Surfaces. 


2.  What  is  the  length  of  an  arc  containing  12°  IC  oi 
12^,"  the  diameter  of  the  circle  being  20  yards  ? 

Ans.  2.1231 
3    What  is  the  length  of  an  arc  of  10°  15'  or  IQio^  in  a 


circle  -whose  diameter  is  68  ? 


Ans.  G.0813. 


PROBLEM     XIII. 

To  find  the  length  of  the  arc  of  a  circle  when  the  chord 
and  radius  are  given. 

RULE. 

1.  Find  the  chord  of  half  the  arc. 

n  From  eight  times  the  chord  of  half  the  arCj  subtract  the 
chord  of  the  whole  arCj  and  divide  the  remainder  by  3,  and  the 
quotient  will  be  the  length  of  the  arc^  nearly. 

EXAMPLES. 

1.  The  chord  AB=20  feet,  and  the 
radius  i4C=20  feet:  what  is  the 
length  of  the  arc  ADB  ? 

First  draw  CD  perpendicular  to  the 
chord  AB :  it  will  bisect  the  chord  at 
P,  and  the  arc  of  the  chord  at  D. 
Then  ^P=  15  feet.     Hence, 

A&-AP=:CP'.  that  is, 

400-225  =  175  and  ^/\^E=U:Z2S=CP 

CD-CP=20-\3.228=6.772=DP. 


Then 
Again, 
hence> 
Then, 


20 


AD=V-JF-KRD^=V'225+45.859984 
AD  1=1 6.4578= chord  of  the  half  arc. 

16.4578x8  —  30 

^ ^=33.8874  =  arc  ADB. 


230  A  P  1'  L  1  C  A  T  I  O  N  S 

jMeusuration    of    Surfaces 

2.  What  is  the  length  of  an  arc  the  chord  of  wliicli  is  21 
feet,  and  the  radius  of  the  circle  20  feet  ? 

Ans.  25.7309  Jt. 

3.  The  cliord  of  an  arc  is  16  and  the  diameter  of  the  ciiclo 
20  :  what  is  the  length  of  the  arc  ?  A?is.  18.5178. 

4.  The  chord   of  an  arc  is   50,  and  the  chord  of  half  the 
arc  is  27  :  what  is  the  length  of  the  arc  ?  Ans.  55 5. 

PROBLEM      XlV. 

To  find  tlie  area  of  a  circle  when  the  diameter  and  circum- 
ference are  both  known. 

RULE. 

Multiply  the  circumference  by  half  the  radius  and  the  product 
will  be  the  area  (Bk.  IV.  Th.  xxvii). 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  10,  and 
circumference  31.416? 
If  the  diameter  be  10,  the 


Operation. 

[6x2i= 
which  is  the  area. 


radius  is  5,  and  half  the  ra- 
dius is  2\ :  hence,  the  cir- 
cumference multiplied  by  2^ 
gives  the  area. 

2.  Find  the  area  of  a  circle  whose  diameter  is  7;  and  cir- 
cumference 21.9912  yards.  Ans.  38.4846  yds. 

3.  How  many  square   yards  in  a  circle  whose  diameter  is 
SJ  feet,  and  circumference  10.9956.  Ans.  1.069016. 

4.  What  is  the  area  of  a  circle  whose  diameter  is  100,  and 
circumference  314.16  ?  Ans  78.')4 


OF      GEOMETRY.  231 

Mensuration    o f    Snrf  ac  e  s. 

6.  What  is  the  area  of  a  circle  whose  diameter  is  I,  and 
circumference  3.1416.  Ans.  0.7854. 

6.  What  is  the  area  of  a  circle  whose  diameter  is  40,  anJ 
circumference  131.9472?  Ans.  1319.472. 

PROBLEM    XV. 

To  find  the  area  of  a  circle  when  the  diameter  only  is 
known. 

RULE. 

Square  the  diameter,  and  then  multiply  by  the  deamal  .7854 

EXAMPLES. 

What  is  the  area  of  a  circle  whose  diameter  is  5  ? 


We  square  the  diameter, 
Tvhich  gives  us  25,  and  we 
then  multiply  this  number 
and  the  decimal  .7854  to- 
gether. 


Operation. 

.7854 
5^=     25 


39270 
15708 


area=  19.6350 


2.  What  is  the  area  of  a  circle  whose  diameter  is  7  ? 

Ans.  38.4846. 

3.  What  is  the  area  of  a  circle  whose  diameter  is  4,5  ? 

.  Ans.  15.90435. 

4.  What  is  the  number  of  square  yards  in  a   iircle  whoso 
diameter  is  1|  yards  ?  Ans.  1.069016. 

5.  What  is  tlie  area  of  a  circle  whose  diameter  is  8.75 
feet?  Ans.  60,1322  sq.ft. 

PROBLEM    XVI. 

To  find  the  area  of  a  circle  when  the  circumference  only 
is  known. 


23?  APPLICATIONS 


Mensuration    of    Surfaces. 


RULE. 

Multiply  the  square  of  the  circumference  by  the  decimal  .07968, 
and  the  product  will  be  the  area  very  nearly 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  circumference  ie 
3.1416? 

Operation. 


We  first  square  the  cir- 
cumference, and  then  multi- 
ply by  the  decimal  .07958. 


3.1416  =9,86965056 
,07958 

area =.7854  + 


2.  What  is  the  area  of  a  circle  whose  circumference  is  91! 

Ans.  659.00198. 

3.  Suppose  a  wheel  turns  twice  in  tracking  16^  feet,  and 
that  it  turns  just  200  times  in  going  round  a  circular  bowling- 
green  :  what  is  the  area  in  acres,  roods,  and  perches  ? 

Ans.  4  A.  3  R.  35.8  P. 

4.  How  many  square  feet  are  there  in  a  circle  whose  cir 
cumference  is  10.9956  yards  ?  Ans.  86.5933. 

5.  How  many  perches  are  there  in  a  circle  whose  circmn 
ference  is  7  miles  ?  Ans.  399300.608. 

PROBLEM      XVII. 

Having  given  a  circle,  to  find  a  square  which  shall  have  an 
equal  area. 

RULE. 

I.  Th6  diameter  X  .S862= side  of  an  equivalent  square 

II.  The  circumference  x  .2821=  side  of  an  equivalent  square 


OF     GEOMETRY.  233 

Mensuration    of    Surfaceg. 

EXAMPLES. 

1.  The  diameter  of  a  circle  is  100:  what  is  the  side  of  a 
square  of  equal  area  ?  Ans.  88. G2. 

2.  The  diameter  of  a  circular  fishpond  is  20  feet,  wlial 
would  be  the  side  of  a  square  fishpond  of  an  equal  area  ? 

Ans.  17.724  ft. 

3.  A  man  has  a  circular  meadow  of  which  the  diameter  is 
875  yards,  and  wishes  to  exchange  it  for  a  square  one  of  equal 
size  :  what  must  be  the  side  of  the  square  ? 

Ans.  775.425. 

4.  The  circumference  of  a  circle  is  200 :  what  is  the  side 
of  a  square  of  an  equal  area  ?  Ans.  56.42. 

5.  The  circumference  of  a  round  fishpond  is  400  yards : 
what  is  the  side  of  a  square  pond  of  equal  area  ? 

Ans.  112.84. 

6.  The  circumference  of  a  circular  bowling-green  is  412 
yards  :  what  is  the  side  of  a  square  one  of  equal  area  ? 

Ans.  116.2252  yds. 

7.  The  circumference  of  a  circular  walk  is  625 :  what  is 
the  side  of  a  square  containing  the  same  area  ? 

Ans.  176.3125. 

PROBLEM      XVIII. 

Having  given  the  diameter  or  circumference  of  a  circle,  to 
find  the  side  of  the  inscribed  square. 

RULE. 

I,  The  diameter  X  .7071  =side  of  the  inscribed  square. 

II.  The  circumference  X  .2251  z^side  of  the  inscribed  square, 

20* 


234  APPLICATIONS 


M  ens  ti  ration    of    Surfaces. 


EXAMPLES. 

1.  Till)  diameter  AB  of  a  circle 
is  400 :  what  is  the  value  of  A  C, 
the  side  of  the  inscribed  square  ?         ^1 

Here, 

.7071  X  400:^282.8400=^0. 


2.  The  diameter  of  a  circle  is  412  feet:  what  is  the  side 
of  the  inscribed  square?  Ans.  291.3252  ft. 

3.  If  the  diameter  of  a  circle  be  600  what  is  the  side  oi 
the  inscribed  square  ?  Ans    424.26. 

4.  The  circumference  of  a  circle  is  312  feet:  what  is  the 
side  of  the  inscribed  square  1  Ans.  70.2312  ft. 

5.  The  circumference  of  a  circle  is  819  yards :  what  is  the 
side  of  the  inscribed  square  ?  Ans.  184.3569  ijds. 

6.  The  circumference  of  a  circle  is  715  :  what  is  the  side 
of  the  inscribed  square  ?  Ans.   J  60.9465, 

7.  The  circumference  of  a  circular  walk  is  625 :  what  is 
the  side  of  an  inscribed  square  ?  Ans.  140.6875. 

TROBLEM     XIX 

To  find  the  area  of  a  circular  sector 

RULE. 

I .  Find  the  length  of  the  arc  by  Problem  XllX 

II,  Multiply  the  arc  by  one  half  the  radius ^  and  the  product 
will  be  the  area 


O  F     G  E  O  M  E  T  R  Y  .  235 


MensuralioD    of    Surfaces. 


EXAMPLES. 

1.  What  is  tho  area  of  the  circular 
sector  ACB,  the  arc  AB  containing 
18°,  and  the  radius  CA  being  equal  to 
3  feet.  i 

First,         .01745  X  18  x3  =  .94230=lengih  AB. 
Then,  .94230  x  ll=1.41345=area 

2.  What  is  the  area  of  a  sector  of  a  circle  in  which  tho  ra- 
dius is  20  and  the  arc  one  of  22  degrees  ? 

Ans.  76.7800. 

3.  Required  the  area  of  a  sector  whose  radius  is  25  and 
the  arc  of  147°  29'.  Ans.  804.2448. 

4.  Required  the  area  of  a  semicircle  in  which  the  radius  is 
13.  ^«^.  2G5.4143. 

5.  What  is  the  area  of  a  circular  sector  when  the  length  of 
the  arc  is  650  feet  and  the  radius  325  ? 

Ans.   105625  sq.  ft. 

PROBLEM      XX. 

To  find  the  area  of  a  segment  of  a  circle. 

RULE. 

I.  Find  the  area  of  the  sector  having  the  same  arc  with  the 
segment,  by  the  last  Problem. 

II.  Find  the  area  of  the  triangle  formed  by  the  chord  of  the 
segment  and  the  two  radii  through  its  extremities. 

Ill  If  the  segment  ts  greater  than  the  semicircle,  add  the  two 
areas  together;  but  if  it  is  less,  subtract  them,  and  the  result  in 
cithei  case,  rmll  be  the  area  required. 


236 


APPLICATIONS 


Mensuration    of    Surfaces 


EXAMPLES. 

1 .  What  is  the  area  of  the  seg- 
ment ADB,  the  chord  AB =24 
feet  and  CA  =20  feet. 


First,       CP=V^-^^ 

=  V'400-144nzl6 
Then, 
PD=CD-CP=20-ie=4. 


And,       ^D=/AP^+PD*=vT444-16  =  12,64911  : 
12,64911x8-24 


then, 


arc  ADB: 


Arc  ADB=25,7309 

half  radius  =  10 


area  sector  ^Z)5C=257,3090 
area  C^5=192 


:25,7309. 

^P=12 
CP=16 

area  C^j5  =  192 


65,309= area  of  segment  ADB 


2.  Find  the  area  of  the  segment 
AFB,  knowing  the  following  lines, 
viz:  ^5=20.5;  PP=  17.17;  AF 
=20;  FG=11.5;  and  C^  =  11.64. 


A       Ar^v    FGxS-AF     11.5x8-20 

Arc  AGF= = =24: 

and  sector      ^GF5C=24x  11.64=279.36  : 
but  CP=PP— ^0=17.17-11.64  =  5.53: 

tI^xGP     20.5x5.53 


Then,  area7lC5=: 


56.6825 


O  F     G  E  O  M  E  T  R  Y  .  23'? 

Mensuration    of    Surfaces. 

Then,  area  of  sector  AFBC=279.3e 

do.  of  triangle  ABC=   56,6825 
gives   area   of  segment   AFB=336.0i25 

3    What  is  the  area  of  a  segment;  the  radius  of  the  circl 
being  10  and  the  chord  of  the  arc  12  yards  ? 

Arts.  16.324  sq.  yds, 

4.  Required  the  area  of  the  segment  of  a  circle  whose 
chord  is  1 6,  and  the  diameter  ol  the  circle  20. 

Ans.  44.5903. 

5.  What  is  the  area  of  a  segment  whose  arc  is  a  quadrant, 
the  diameter  of  the  circle  being  18?  Ans.  63.6174. 

6.  The  diameter  of  a  circle  is  100,  and  the  chord  of  the 
seo-ment  60  :   what  is  the  area  of  the  segment  ? 

Ans.  408,  nearly. 

PROBLEM    XXI. 

To  find  the  area  of  an  ellipse. 

Multiply  the  two  axes  trgethcr,  and  their  product  by  the  decimal 
,7854,  and  the  result  will  be  the  required  area. 

EXAMPLES. 

1.  Required  the  area  of  an  ellipse, 
whose  transverse  axis  AB=ilO  feet, 
and  the  conjugate  axis  DE =50  feet. 

ABxDE=70x50=3500: 

Then,  .7854  x  3500 =2748.9= area. 

2.  Required  the  area  of  an  ellipse  whose  axes  are  24  and 
t  \  Ans.  339.2928. 


238  APPLICATIONS 

Mensuration    of    Surfaces, 

3.  What  is  the  area  of  an  ellipse  whose  axes  are  80  and 
60  ?  Ans.  3769.92. 

4.  What  is  the  area  of  an  ellipse  whose  axes  are  50  and 
1?  Ans.  1767.15. 

PROBLEM    XXII. 

To  find  the  area  of  a  circular  ring :  that  is,  the  area  in- 
cluded between  the  circumferences  of  two  circles,  having  a 
common  centre. 

RULE. 

I.  Square  the  diameter  of  each  ring,  and  subtract  *he  square 
of  the  less  from  that  of  the  greater. 

II.  Multiply  the  difference  of  the  squares  by  thtr  d'cimai 
7854,  and  the  product  will  be  the  area. 

EXAMPLES. 


1.  In  the  concentric  circles 
having  the  common  centre  C,  we 
have 

^5  =  10  yds.,  and  DE  =  6  yards  : 
what  is  the  area  of  the  space  in- 
cluded between  them  ? 


5.4*=10*=100 
DE^=   ?=   36 


Difference =64 
Then,  63  X  .7854  =  50,2656=:  area. 

2.  What  is  the  area  of  the  ring  when  the  diameters  of  the 
circle  are  20  and  10  f  Ans.  235.62. 


OF     GEOMETRY.  239 

Mensuration    of    Solids. 

3.  If  the  diameters  are  20  and  15,  what  will  be  the  area  in- 
cluded between  the  circumferences  ?  Ans    137.445. 

4.  If  the  diameters  are  IG  and  10,  what  will  be  the  area  ia- 
cluded  between  the  circumferences  ?  Ans.  122.5224. 

5    Two  diameters  are  21.75  and  9.5  ;  required  the  area  o( 
the  circular  ring.  Ans.  300.6600 

6.  If  the  two  diameters  are  4  and  6,  what  is  ihe  area  of  the 
rin^?  Ans.  15.708 


MENSURATION      OF      SOLIDS. 


DEFINITIONS. 


The  mensuration  of  solids  is  divided  into  two  parts. 

Ist,  The  mensuration  of  the  surfaces  of  solids  :  and 

2d,  The  mensuration  of  their  solidities. 

We  have  already  seen  that  the  unit  of  measure  for  plane 
surfaces,  is  a  square  whose  side  is  the  unit  of  length  (Bk.  IV 
Dcf.  7). 

2.  A  curve  line  which  is  expressed  by  numbers  is  also  re- 
ferred to  an  unit  of  length,  and  its  numerical  value  is  the  num- 
ber of  times  which  the  line  contains  the  unit. 

If  then,  we  suppose  the  linear  unit  to  be  reduced  to  a 
Btiftight  line,  and  a  square  constructed  on  this  line,  this  square 
will  be  the  unit  of  measure  for  curved  surfaces. 

3.  The  unit  of  solidity  is  a  cubo,  whose  edge  is  the  unit  in 
which  the  linear  dimensions  of  the  solid  are  expressed ;  and 


240 


APPLICATIONS 


Mensuration    of    Solids. 


the  face  of  this  cube  is  the  superficial  unit  in  which  the  but- 
fiace  of  the  solid  is  estimated  (Bk.  VI.  Th.  xiii.  Sch). 

4    The  following  is  a  table  of  solid  measure. 

1  cubic  foot    =1728        cubic  inches. 
1   cubic  yard   =27 
1   cubic  rod     =4492^ 
1  ale  gallon    =282 
1  wine  gallon =231 


1   bushel 


cubic  feet, 
cubic  feet, 
cubic  inches, 
cubic  inches. 
:2150,42  cubic  inches. 


PROBLEM      1. 

To  find  the  surface  of  a  right  prism. 

RULE. 

Multiply'  the  perimeter  of  the  base  by  the  altitude  and  the  pro- 
duct will  be  the  convex  surface :  and  to  this  add  the  area  of  the 
bases  J  when  the  entire  surface  is  required  (Bk.  VI.  Th.  i). 

EXAMPLES 


1.  Find  the  entire  surface  of  the 
rcgidar  prism  whose  base  is  the  reg- 
ular polygon  ABCDE  and  altitude 
AFf  when  each  side  of  the  base  is 
20  feet  and  the  altitude  AF,  50  feet. 


/>~ 


1) 


B  C 

A84-5C-hCD+D£4-E^  =  100;  and  ^i^=50 :  then 
(AB-\-BC->t-  CD-\-DE-^EA)  x  ^F=convex  surface 


OF     GEOMETRY.  24i 


Mensuration    of    Solida 


which  becomes,  100x50  =  5000  square  feet;  which  is  ihe 
convex  surface.     For  the  area  of  the  end,  we  have 
AB  X  tabular  number = area -A5Ci)i?, 
that  is,  2?*  X  tabular  number,  or  400  x  1.720477  =  688.1908^ 
the  area  ABODE. 

Then,  convex  surface  =  5000  square  feet, 

lowei  base  688.1908    square  feet, 

upper  base  688.1908    square  feet. 

Entire  surface  6376^816 

2.  What  is  the  surface  of  a  cube,  the  length  of  each  sme 
being  20  feet  ?  Ans.  2400  sq.  ft. 

3.  Find  the  entire  surface  of  a  triangular  prism,  whose  base 
is  an  equilateral  triangle,  having  each  of  its  sides  equal  to  18 
inches,  and  aliiiude  20  feet.  Ans.  91.949  sq.  ft. 

4.  What  is  the  convex  su'-face  of  a  regular  octagonal  prism, 
^he  side  of  whose  base  is  15  and  altitude  12  feet? 

Ans.    1440  sq.  ft. 

5.  What  must  be  paid  for  lining  a  rectangular  cistern  w'th 
lead  at  2d  a  pound,  the  thickness  of  the  lead  being  such  as  to 
require  lib.  for  each  square  foot  of  surface  ;  the  inner  dimen- 
sions of  the  cistern  being  as  follows :  viz.  the  length  3  feet  2 
inches,  the  breadth  2  feel  8  inches,  and  the  depth  2  feet  6 
inches  ?  Ans.  £2  3s     lOf^f. 

PROBLEM      II 

To  find  the  solidity  of  a  prism. 

RULE. 

Multiply  the  area  of  the  base  by  the  perpendicular  height,  and 
the  product  will  he  the  solidity. 


242 


APPLICATIONS 


Mensurat'on    of    Sol'ds 


BX  AMPLK8. 


1.  What  is  the  solidity  of  a  reg- 
ular pentagonal  prism  vvliose  altitude 
IS  20,  and  each  side  of  the  base  15 
feet? 

To  find  the  area  of  the  base  we 
have  by  Problem  VIII,  page  178. 


.>\ 


15^^=225:  and  225x1.7204774  =  387.107415=. 

the  area  of  the  base  :  hence, 

387,107415  X20  =  7742.1483  =  solidity. 

2.  What  is  the  solid  cimtents  of  a  cube  whose  side  is  is-l 
inches  ?  Ans.  13824  solid  in. 

3.  How  many  cubic  feet  in  a  block  of  marble,  of  which  tlie 
length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and  height 
or  thickness  2  feet  6  inches  ?.  Ans.  2H  solid  ft. 

4.  How  many  gallons  of  water,  ale  measure,  will  a  cistern 
contain  whose  dimensions  are  the  same  as  in  the  last  ex- 
ample ?  Ans.   1291^ 

5.  Required  the  solidity  of  a  triangular  prism  ^hose  alti- 
tude is  10  feet,  and  the  three  sides  of  its  triangular  base  3,  4, 
and  6  feet.  Ans.  60  solid  ft. 

0.  What  is  the  solidity  of  a  square  prism  whoce  heiglit  ii> 
6{  feet,  and  each  side  of  the  base  1^  fcot? 

Ans    9|  soha  ft. 


OF     GEOMETRY 


243 


Mensuration    of    Solids. 


7.  What  is  the  sohdity  of  a  prism  whose  base  is  an  equi- 
lateral triangle,  each  sitle  of  which  is  4  feet,  the  height  of  the 
prism  being  10  feet?  Ans.  G9.282  solid  ft. 

8.  Wliat  is  the  number  of  cubic  or  solid  feel  in  a  resulai 
penti.gonal  prism  of  which  the  altitude  is  15  feet  and  each 
Rifle  of  the  base  3.75  feet  ?  Ans.  362.913 

PROBLEM    III. 

To  find  the  surface  of  a  regular  pyramid. 

RULE. 

Multiply  the  perimeter  of  the  base  by  half  the  slant  lieight^ 
and  the  product  will  be  the  convex  surface :  to  this  add  the  area 
of  the  base,  if  the  entire  surface  is  required  ifik.  VI.  Th  \\\ 


EXAMPLES. 

1.  In  the  regular  pentagonal  pyramid 
S—ABCDE,  the  slant  height  SF  is 
equal  to  45,  and  each  side  of  the  base 
IS  15  feet :  required  the  convex  sur- 
face, and  also  the  entire  surface. 

15  X5  =  75  =  perimeter  of  the  base, 
75x22^  =  1087.5  square  feet  =  area  of 
ctmvex  surface. 


And   15^=225:  then 225  x  1.7204774  =  387.1074151^ the  area 

of  the  base. 

Ilcnce,         convex  surface   =1687.5 

area  of  the  base=  387.107415 
Entire  surface  =2074.607415  square  feet 


'M4 


APPLICA.TIONS 


Mensuration    of    Solids 


2.  What  is  the  convex  surface  of  a  regular  triangular  pyra- 
mid, the  slant  height  being  20  feet,  and  each  side  of  the  base 
3  fed  ?  Ans.  90  sq.  ft 

3.  What  is  the  entire  surface  of  a  regular  pyramid  whose 
ftiaiit  height  is  15  feet,  and  the  base  a  regular  pentagon,  ol 
which  each  side  is  25  feel?  Ans.  2012.798  sq.  ft 


PROBLEM    IV. 

To  find  the  convex  surface  of  the  frustum  of  a  regidai 
pjTamid. 

RULE. 

Multiply  half  the  sum  of  the  perimeters  of  the  two  bases  by 
the  slant  height  of  the  frustum,  and  the  product  will  be  the  con' 
vex  surface  (Bk.  VI.  Th.  vii). 

EXAMPLES. 

1.  In  the  frustum  of  the  regular  pen- 
tagonal pyramid  each  side  of  the  lovi^er 
base  is  30,  and  each  side  of  the  upper 
base  is  20  feet,  and  the  slant  height 
fF  is  equal  to  15  feet.  What  is  the 
convex  surface  of  the  frustum  ? 

Ans.  1875  sq.  ft. 

2.  How  many  square  feet  are  there  in  the  convex  surface 
of  the  frustum  of  a  square  pyramid,  whose  slant  height  is  10 
feot,  each  side  of  the  lower  base  3  feet  4  inches,  and  each 
evif  of  the  upper  base  2  feet  2  'nches  '  Ans.  1 10. 

3.  What  is  the  convex  surface  oi  the  frustum  of  a  heptago 

nai  pjTamid  whose  slant  height  is  55  feet,  each  side  of  the 

lowoi  base  8  feet,  and  each  side  of  the  upper  base  4  feet  ? 

Ans.  2310  sq.  ft. 


OF     GEOMETRY 


245 


Mensuration    ol     Solids. 


PROBLEM    V. 

To  find  the  solidity  of  a  pjTamid. 


RULE. 


Multiply  the  area  of  the  base  hy  the  altitude  and  divide  the  pro- 
duct hy  3,  the  quotient  will  he  the  solidity  (Bk.  VI.  Tli.  xvii). 


FXAMPLES. 


1  What  is  the  solidity  of  a  pyramid 
Uie  area  of  whose  base  is  215  square 
feet  and  the  altitude  50=45  feet? 

First,         215x45  =  9675: 

then,        9675  H-   3  =  3225 
which  is  the  solidity  expressed  in  solid 
feet. 


2.  Required  the  solidity  of  a  square  pyramid,  each  side  ol 
its  base  being  30  and  its  altitude  25.        Ans.  7500  solid  ft. 

3.  How  many  solid  yards  are  there  in  a  triangular  pyramid 
whose  altitude  is  90  feet,  and  each  side  of  its  base  3  yards  ? 

Ans.  38.97117. 

4.  How  many  solid  feet  in  a  triangular  pyramid  the  altitude 
of  which  is  14  feet  6  inches,  and  the  three  sides  of  its  base  5, 
6  and  7  feel?  Ans.  71.0352. 

5.  What  is  the  solidity  of  a  regular  pentagonal  pyramid,  its 

altitude  being  12  feet,  and  each  side  of  its  base  2  feet  • 

Ans  27.527G  solid  ft, 
21* 


5240  APPLICATIONS 


Mensuration    of    Solids, 


6  How  many  solid  feet  in  a  regular  hexagonal  pyramid 
whose  altitude  is  6.4  feet,  and  each  side  of  the  base  6  inches ' 

Ans.  L385G4. 

7.  How  many  solid  feet  are  contained  in  a  hexagonal  p}T:a- 
mid  the  height  of  which  is  45  feet,  and  each  side  of  the  base 
10  feet?  Ans.  3897.1143. 

8.  The  spire  of  a  church  is  an  octagonal  pyramid,  each  side 
of  the  base  being  5  feet  10  inches,  and  its  perpendicular 
height  45  feet.  Within  is  a  cavity,  or  hollow  part,  each  side 
of  the  base  being  4  feet  1 1  inches,  and  its  perpendicular 
height  41  feet:  how  many  yards  of  stone  does  the  spire 
contain'  Ans.  32.197353 

PROBLEM    VI. 

To  hnd  the  solidity  of  the  frustum  of  a  pyramid. 

RULE. 

Add  together  the  areas  of  the  two  bases  of  the  frustum  and 
a  geometrical  mean  vroportional  between  them ;  and  then  multi- 
ply the  sum  by  the  altitude,  and  take  one-third  the  product  for 
the  solidity. 

EXAMPLES. 


1.  What  is  the  solidity  of  the  frus- 
tum of  a  pentagonal  pyramid  the  area 
of  the  lower  base  being  16  and  of  the 
upper  base  9  square  feet,  the  altitude 
being  1  feet ' 


O  F     G  E  O  M  E  T  R  Y  .  24' 


Mensuration    of    Solids 


First,  16x9=144:  then,-v/l'^'*  =  12,  the  mean 
Then,  area  of  lower  base  =16 

area  of  upper  base   =  9 

mean  of  bases  =  12 

height  7 

3)  259 
solidity  =86^  solid  ft. 

2.  What  is  the  number  of  solid  feet  in  a  piece  of  timbei 
whose  bases  are  squares,  each  side  of  the  lower  base  being 
15  inches,  and  each  side  of  the  upper  base  being  6  inches, 
the  length  being  24  feet?  Ans.  19.5. 

3.  Required  the  solidity  of  a  regular  pentagonal  frustum, 
whose  altitude  is  5  feet,  each  side  of  the  lower  base  18 
inches,  and  each  side  of  the  upper  base  6  inches. 

Ans.  9.31925  solid  ft. 

4.  What  is  the  contents  of  a  regular  hexagonal  frustum, 
whose  height  is  6  feet,  the  side  of  the  greater  end  18  inches, 
and  of  the  less  end  12  inches?       Ans.  24.681724  cubic  ft. 

5.  How  many  cubic  feet  in  a  square  piece  of  timber,  the 
areas  of  the  two  ends  being  504  and  372  inches,  and  its 
length  3U  feet  ?  Ans.  95.447. 

6.  What  is  the  solidity  of  a  squared  piece  of  timber,  its 
length  being  18  feet,  each  side  of  the  greater  base  18  inchcfl, 
and  each  side  of  the  smaller  12  inches  ? 

Ans.  28.5  aibte  ft. 

7.  What  is  the  solidity  of  the  frustum  of  a  regular  hexago- 
tial  pyramid,  the  side  of  the  greater  end  being  3  feet,  that  of 
the  less  2  feet,  and  the  height  12  feet? 

Ans.  197.453776  solid  ft 


248 


APPLICATIONS 


Mensuration    of    Soli  do 


MEASURES     OF     THE      THREE     ROUND     BODIES. 
PROBLEM     I 

To  find  the  surface  of  a  cylinder. 

RULE. 

Multiply  the  circumference  of  the  base  by  the  altitude,  and  the 
product  will  be  the  convex  surface  ;  and  to  this,  add  the  areas  of 
the  two  bases,  when  the  entire  surface  is  required  (Bk.  VI.  Th.  iiV 


EXAMPLES. 

1.  What  is  the  entire  surface  of  the 

cyhnder  in  which  AB,  the   diameter  of 

the  base,  is  12  feet,  and  the  aUitude  EF 

30  feet  ? 

First,  to  find  the  circumference  of  the 

base,  (Prob.  X.  page  180) :  we  have 

3.1416  X  12  =  37.6992  =  circumference  of 

the  base. 

Then,  37.6992  X  30=  1130.9760=convex  surface. 

Also,  12^=144:     and     144  x  .7854  =  113.0976  =  area 

base. 

Then.  convex  surface  =1130.9760 

lower  base  113.0976 

upper  base  113.0976 

Entire  area    =1357.1712 


of  the 


2.  What  is  the  convex  curface  of  a  cylinder,  the  diameter 
of  whose  base  is  20,  and  the  altitude  50  feet  ? 

Ans    3141.6  sq.ft. 


OF      GEOMETRY 


249 


Mensuration    of    the    Round    Bodies. 

3.  Required  the  entire  surface  of  a  cylinder,  whose  altitude 
is  20  feet  and  the  diameter  of  the  base  2  feet. 

Ans.  131.9472 /^ 

4.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  30  inches,  and  altitude  5  feet? 

Ans.  5654.88  sq.  in. 

5.  Required  the  convex  surface  of  a  cylinder,  whose  aJti- 
tude  is  14  feet,  and  the  circumference  of  the  base  8  feet  i 
inches.  Ans.  116.6666,  <fec.,  ^y.  ft. 


PROBLEM     II. 

To  find  the  solidity  of  a  cylmder. 

RULE. 

Multiply  the  area  of  the  base  by  the  altitude,  and  the  prodtu 
will  be  the  solidity. 

EXAMPLES. 

1.  What  is  the  solidity  of  a  cylinder, 
the  diameter  of  whose  base  is  40  feet, 
and  altitude  EFy  25  feet  ? 

First,  to  find  the  area  of  the  base,  we 
have  (Prob.  xv.  page  231). 

40'=1600:   then,  1 600 X. 7854  =1256.64. 

=area  of  the  base. 

Then,  1256.64x25=31416  solid  feet,  which  is  the  solidity. 

2.  What  is  the  solidity  of  a  cylinder,  the  diameter  of  whose 
base  is  30  feet,  and  altitude  50  feet  ? 

Ans    35343    cubic    ft. 


250  APPLICATIONS 


Mensuration    of     the    Round    Bodies. 


3.  WTial  is  the  solidity  ol  a  cylinder  whose  height  is  5  feet, 
and  the  diameter  of  the  end  2  feet?       Ans.  15.708  solid  ft. 

4.  What  is  the  solidity  of  a  cylinder  whose  height  is  20 
feet,  and  the  circumference  of  the  base  20  feet  ? 

Ans.  636.64  atOic  ft 

5.  The  circumference  of  the  base  of  a  cylinder  is  20  feet, 
and  the  altitude  19.318  feet':  what  is  the  solidity? 

Ans.  614.93  cubic  ft. 

6.  What  is  the  solidity  of  a  cylinder  whose  altitude  is  12 
feet,  and  the  diameter  of  its  base  15  feet  ? 

Ans.  2120.58  cubic  ft. 

7.  Required  the  solidity  of  a  cylinder  whose  altitude  is  20 
feet,  and  the  circumference  of  whose  base  is  5  feet  6  inches ' 

Ans.  48.1459  cubic  ft. 

8.  What  is  the  solidity  of  a  cylinder,  the  circumference  of 
whose  base  is  38  feet,  and  altitude  25  feet  ? 

Ans.  2872.838  cubic  ft. 

9.  What  is  the  solidity  of  a  cylinder,  the  circumference  of 
whose  base  is  40  feet,  and  altitude  30  feet  ? 

10.  The  diameter  of  the  base  of  a  cylinder  is  84  yards,  and 
the  altitude  21  feet :  how  many  solid  or  cubic  yards  does  it 
contain  ?  Ans.  38792.4768. 

PROBLEM      III 

To  find  the  siurface  of  a  cone. 

RULE. 

Multiply  the  circumference  of  the  base  by  the  slant  height,  and 
divide  the  product  by  2  ;  the  quotient  will  be  the  convex  surface, 
to  which  add  the  area  of  the  base,  when  the  entire  surface  in 
required  (Bk.  VI.  Th.  viii) 


OF      GEOMETRY 


251 


Mensuration    of    the    Round    Bodio9 


EXAMPLES. 


1.  What  IS  the  convex  surface  of  the 
cone  whose  vertex  is  C,  the  diameter 
ADi  of  its  base  being  8|  feet,  and  the 
side  CA,  50  feet. 


First,     3.1416  X  8^= 26.7036 =circumference  of  base 

26.7036X50 
Ihen = 007.59  rr convex  surface. 


2.  Required  the  entire  surface  of  a  cone  whose  side  is  36 
and  the  diameter  of  its  base  18  feet. 

Ans.   1272.348  sq.  ft. 

3.  The  diameter  of  the  base  is  3  feet,  and  the  slant  height 
15  feet :  what  is  the  convex  surface  of  the  cone  ? 

Ans.  70.686  sq.  ft. 

4.  The  diameter  of  the  base  of  a  cone  is  4,5  feet,  and  the 
slant  height  20  feet :   what  is  the  entire  surface  ? 

Ans.  157.27G35  sq.  Jt. 

5.  The  circumference  of  the  base  of  a  cone  is  10.'''5,  and 
the  slant  height  is  18.25  :  what  is  the  entire  surface  ? 

Ans.  107.29021  sq.  ft 

PROBLEM     IV. 

To  find  the  solidity  of  a  cone. 

RULE. 

Multiply  the  area  of  the  base  by  the  altitude;  and  divide  the  prv 
duct  by  3,  the  quotient  will  be  the  solidity  (Bk.  VI.  Th.  rdii). 


252 


APPLICATIONS 


Mensuration    of    the    Round    Bodies. 


EXAMPLES. 


1.  What  is  the  solidity  of  a  cone,  the 
area  of  whose  base  is  380  square  feet, 
and  altitude  CBy  48  feet  ? 


We  simply  multiply  the 
area  of  the  base  by  the  alti- 
tude, and  then  divide  the  pro- 
duct by  3. 


3040 
1520 
3)18240 
area =6080 


2.  Required  the  solidity  of  a  cone  whose  altitude  i?  21 
feet,  and  the  diameter  of  the  base  10  feet. 

Ans.  706.86  cubic  ft. 

3.  Required  the  solidity  of  a  cone  whose  altitude  is  1 04 
feet,  and  the  circumference  of  its  base  9  feet  ? 

Ans.  22.5609  cubic  fi. 

4.  What  is  the  solidity  of  a  cone,  the  diameter  of  whose 
base  is  18  inches,  and  altitude  15  feet? 

Ans.  8,83575  cubic  ft. 

5    The  circumference  of  the  base  of  a  cone  is  40  feet,  and 
ihe  altitude  50  feet :  what  is  the  solidity  1 

An*.  2122.1333  solid  fi. 


OF     GEOMETRY 


253 


Mensuration    of    the    Round    Bodiea. 

PROBLEM     v. 

To  find  the  surface  of  the  frustum  of  a  cone 

RULE. 

Add  together  the  circumferences  of  the  two  bases;  and  multi' 
^ly  the  sum  by  half  the  slant  height  of  the  frustum;  the  product 
will  be  the  convex  surface,  to  which  add  the  areas  of  the  bases 
when  the  entire  surface  is  required  (Bk.  VI.  Th.  ix). 

EXAxMPLES. 


1.  What  is  tlie  convex  surface  of  the 
frustum  of  a  cone,  of  which  the  slant 
height  is  12|  feet,  and  the  circumfe- 
rences of  the  bases  8,4  and  6  feet. 


We  merely  take  the  sum 
of  the  circumferences  of  the 
bases,  and  multiply  by  half 
the  slant  height,  or  side. 


Operation. 

8.4 

6 

14.4 

half  side     6.25 


area =90  sq.  ft. 

2.  What  is  the  entire  surface  of  the  frustum  of  a  cone,  the 
side  being  16  feet,  and  the  radii  of  the  bases  2  and  3  feet  ? 

Ans.  292.1688  sq.  ft. 

3.  What  is  the  convex  surface  of  the  frustum  of  a  cone, 
iho  circumference  of  the  greater  base  being  30  feet,  and  of 
&o  less  10  feet;  the  slaut  height  being  20  feet? 

Ans.  400  sq.  ft. 

4.  Required  the  entire  surface  of  the  frustum  of  a  cone 

whose  slant  height  is  20  feet,  and  the  diameters  of  the  basc^ 

8  and  4  feet  Ans.  439.824  sq.  ft. 

22 


254 


APPLICATIONS 


Mensuration    of    the    Round    Bodies 


PROBLEM    VI. 

To  find  the  solidity  of  the  frustum  of  a  cone 

RULE. 

I.  Add  together  the  areas  of  the  two  ends  and  a  geometrical 
mean  between  them 

II.  Multiply  this  sum  by  one-third  of  the  altitude  and  the 
product  will  be  the  solidity. 

EXAMPLES. 

1  How  many  cubic  feet  in  the  frus- 
tum of  a  cone  whose  altitude  is  26  feet, 
and  the  diameters  of  the  bases  22  and 
1 8  feet  ? 

First,   22^ X. 7854=: 380. 134=^ area    of 
ower  base : 
and     18^  X  .7854=r254.47  =  area  of  upper  base 


Then,  V380.134x254.47=311.018=mean. 

26 
Then,  (380.134+254,47+311.018)  x— -^8195.39    which 

o 

is  the  solidity. 

2.  How  many  cubic  feet  in  a  piece  of  round  timber  the  di- 
ameter of  the  greater  end  being  18  inches,  and  that  of  the  less 
9  inches,  and  the  length  14.25  feet  ?  An^.  14.68943. 

3.  What  is  the  solidity  of  a  frustum,  the  altitude  beiiig  18. 
tho  diameter  of  the  lower  base  8,  and  of  the  upper  4  ? 

Ans.  527.7888. 

4.  If  a  cask,  which  is  composed  of  two  equal  conic  frus- 
tums joined  together  at  their  larger  bases,  have  its  bung  di- 
ameter 28  inches,  the  head  diameter  20  inches,  and  the  len^h 


OF     GEOMETRY. 


255 


Mensuration    of    the    Round    Bodies. 

40  inches,  how  many  gallons  of  wine  will  it  contain,  there 
being  231  cubic  inches  in  a  gallon  ?  Ans.  79.0613. 

PROBLEM     VII. 

To  find  the  surface  of  a  sphere. 

RULE. 

Multiply  the  circumference  of  a  great  circle  hy  the  diameter ,  and 
the  product  will  be  the  surface  (Bk.  VI.  Th.  xxiii). 

EXAMPLES. 


1 .  What  is  the  surface  of  the  sphere 
whose  centre  i.s  C,  the  diameter  being 
Tfeot? 

Ans.  153.9384  sq.  ft. 


2.  What  is  the  surface  of  a  sphere  whose  diameter  is  24  1 

Ans.  1809.561G. 

3.  Required  the  surface  of  a  sphere  whose  diameter  is 
7921  miles.  Ans.  197111024  sq.  miles. 

\ .  What  is  the  surface  of  a  sphere  the  circumference  o( 
whose  great  circle  is  78.54  ?  Ans.  1963.5. 

5.  What  is  the  surface  of  a  sphere  whose  diameter  .s  ..  j 
feet  ?  Afis.  5.58506  sq.  ft, 

PROBLEM     VIII. 

To  find  the  convex  surface  of  a  spherical  zone. 

RULE. 

Multiply  the  height  of  the  zone  by  the  circumference  of  a  great 
circle  of  the  sphere,  and  the  product  will  be  the  convex  surface 
(Bk.  VI.  Th.  xxiv) 


256 


APPLICATIONS 


Mfnsuration    of    the    Round    Bodies, 


EXAMPLES. 


1.  What  is  the  convex  surface  of 
the  zone  ABD,  the  height  BE  being 
9  inches,  and  the  diameter  of  the 
sphere  42  inches  ? 


First,  42  X  3.1416  =  1 31 .9472  =  circumference, 

height  =  9 

surface  =11 87.5248  square  inches. 

2.  The   diameter  of  a  sphere  is   12|  feet :  what  will  be 
the  surface  of  a  zone  whose  altitude  is  2  feet  ? 

Ans.  78.54  sq.  ft. 

3.  The  diameter  of  a  sphere  is  21  inches  :  what  is  the  sur- 
face of  a  zone  whose  height  is  4^  inches  ? 

Ans.  296.8812  sq.  in. 

4.  The  diameter  of  a  sphere  is  25  feet  and  the  height  o/ 
the  zone  4  feet :  what  is  the  surface  of  the  zone  ? 

Ans.  314.16  5^.  ft. 

5.  The  diameter  of  a  sphere  is  9,  and  the  height  of  a  zone 
3  foet :  what  is  the  surface  oi  the  zone  ? 

Ans.  84.8232. 

PROBLEM    IX. 

To  find  the  solidity  of  a  sphere. 


RULE 


Multiply  the  surface  hy  one-third  of  the  radius  and  the  product 
will  be  the  solidity  (Bk.  VI.  Th.  xxv)- 


OF     GEOMETRY 


251 


Mensuration    of    the    Roiind    Boiias. 


EXAMPLES. 

1.  What  is  ihe  solidity  of  a  sphere 
whcec  diameter  is  12  feet? 

First,         3.1416x12  =  37.0992  = 
circumference  of  sphere. 

diameter  =  12 

surface  =452.3904 

one^hird  radius         =  2 

Solidity  =904.7808  cubic  feet. 

2.  The  diameter  of  a  sphere  is  7957.8:  wliat  is  its  solidity? 

Ans.  263863122758.4778. 

3.  The  diameter  of  a  sphere  is  24  yards  :  what  is  its  solid 
contents  ?  Ans.  7238.2464  cubic  yds. 

4.  The  diameter  of  a  sphere  is  8  :  what  is  its  solidity? 

Ans.  268.0832. 

A    The  diameter  of  a  sphere  is  16  ;  what  is  its  solidity  ? 

Ajis.  2144.6656 

RULE    11. 

Citde  the  diameter  and  multiple/  the  number  thus  founds  by  th4 
decimal  .5236,  and  the  product  will  be  the  solidity. 


EXAMPLES. 

I    What  is  the  solidity  of  a  sphere  whose  diameter  is  20  ? 

Ans.  4188.8. 

2.  What  is  the  solidity  of  a  sphere  whose  diameter  is  6? 

Ans.  113.0976. 

3.  What  is  the  solidity  of  a  sphere  whose  diameter  is  10 : 

2^^  Ans    .'=^23.6 


258 


APPLICATIONS 


Me  nsu  ration    of    the    Round    Bodies. 

PROBLEM    X. 
To  find  the  solidity  of  a  spherical  segment  with  one  base. 

RULE. 

I.  71  three  times  the  square  of  the  radius  of  the  base,  add  the 
iquare  of  the  height. 

II.  Multiply  this  sum  by  the  height,  and  the  product  by  the 
decimal  .5236,  the  result  will  he  the  solidity  of  the  segment. 

EXAMPLES. 

1 .  What  is  the  solidity  of  the  seg- 
ment ABDj  the  height  BE  being  4 
feet,  and  the  diameter  AD  of  the 
base  being  14  feet? 

First, 

?x3-f  4^=1474-16  =  163: 

Then,  163 x4x. 5236  =  341.3872  solid  feet,  which  is  the 
solidity  of  the  segment. 

2.  What  is  the  solidity  of  the  segment  of  a  sphere  whose 
neight  is  4,  and  the  radius  of  its  base  8  ?        Ans.  435.6352. 

3.  What  is  the  solidity  of  a  spherical  segment,  the  diam- 
eti^r  of  its  base  being  17.23368,  and  its  height  4.5  ? 

Ans.  572.5566. 

4-  What  is  the  solidity  of  a  spherical  segment,  the  diam- 
eter of  the  sphere  being  8,  and  the  height  of  the  segment  2 
fett  ?  Ans.  4 1.888  cubic  Jt. 

5  What  is  the  solidity  of  a  segment,  when  the  diameter 
of  the  sphere  is  20,  and  the  altitude  of  the  segment  9  feet  ? 

Ans.  1781.2872  cubic  ft 


OF      G  E  0  I\I  E  T  R  Y 


259 


Mensuration    of    the    Spheioid 


OF     THE     SPHEROID. 

A  spheroiil  is  a  solid  described  by  the  revolution  of  an 
ellipse  about  either  of  its  axes. 

If  an  ellipse  ACBD^  be  re- 
volved about  the  transverse  or 
onger  axis  AB,  the  solid  de- 
scribed is  called  a  prolate 
spheroid :  and  if  it  be  revolved 
about  the  shorter  axis  CZ),  the  solid  described  is  called  an 
oblate  spheroid. 

The  earth  is  an  oblate  spheroid,  the  axis  about  wliich  it 
revolves  being  about  34  miles  shorter  than  the  diameter  per- 
|)endicular  to  it. 

PROBLEM    XI. 

To  find  the  solidity  of  an  ellipsoid 

RULE. 

Multiply  the  fixed  axis  by  the  square  of  ths  revolving  axu, 
and  the  product  by  the  decimal  .5236,  the  result  will  be  the  re- 
quired solidity. 

EXAMPLES. 

1.  In  the  prolate  spheroid 
ACBDj  the  transverse  axis 
AB=i90,  and  the  revolving 
axis  CD  =  70  feet:  what  is 
the  solidity? 

Here,  ^5=90  feet:   OD^  =z 70^=4900  :  hence 
AB  X  CD^  X  .5236=90  x  4900  x  .5236=230907.6  cubic  feet, 
vvhich  is  the  solidity. 


2G0 


APPLICATIONS 


Mensuration    of    Cylindrical    R  ing9. 

2.  What  is  the  solidity  of  a  prolate  spheriod,  whoso  fixed 
axisis  100,  and  revolving  axis  6  feet?  Ans.  1884.96. 

3.  What  is  the  solidity  of  an  oblate  spheroid,  whose  fixrJ 
axis  is  60,  and  revolving  axis  100  ?  Ans.  314160. 

4.  What  is  the  solidity  of  a  prolate   spheroid,  whose  axes 
are  40  and  50  ?  Afis.  41888. 

5.  What  is  the  solidity  of  an  oblate  spheroid,  whose  axee 
are  20  and  10?  Ans.  2094.4. 

6.  What  is  the  solidity  of  a  prolate  spheroid,  whose  axes 
are  55  and  33?  Ans.  31361.022. 

7.  What  is  the  solidity  of  an  oblate  spheroid,  whose  axes 
are  85  and  75  ?  Ans.  


OF      CYLINDRICAL      RINGS 

A  cylindrical  ring  is  formed  by 
bending  a  cylinder  until  the  two 
ends  meet  each  other  Thus,  if  a 
cylinder  be  bent  round  until  the  axis 
takes  the  position  mon,  a  solid  will 
be  formed,  which  is  called  a  cylin- 
drical ring. 

The  line  AB  is^  called  the  outer,  and  cd  the  inner  diameter. 

PROBLEM     XII. 

To  find  the  convex  surface  of  a  cylindrical  ring. 


RULE. 


I.  To  the  thickness  of  the  ring  add  the  inner  diameter. 

II.  Multiply  this  sum  hy  the  thickness,  and   the  p'oduci   by 
9,8696,  the  result  will  be  the  area. 


O  F      G  E  0  M  E  T  R  Y  26 1 

Mensuration    of    Cylindrical    Ri  ngs. 

EXAMPLES. 

1 .  The  thickness  Ac,  of  a  cylindri- 
cal ring  is  3  inches,  and  ilie  inner 
diameter  c</,  is  12  inches:  what  is 
(lie  convex  snrface  ? 

I  Ac-\-cd=^3  \-VZ  =  \b:  %    \^ 

15x3x9.8G96=:444.132  square 
inches  =  the  surface. 

2.  The  thickness  of  a  cylindrical  ring  is  4  inches,  and  the 
inner  diameter  18  inches  :  what  is  the  convex  surface  ? 

Ans.  868.52  sq.  in. 

3.  The  thickness  of  a  cylindrical  ring  is  2  inches,  and  the 
iniier  diameter  18  inches  •  what  is  the  convex  surface  ? 

Ans.  391.784  sq.  in. 
PROBLEM    XIII. 

To  find  the  solidity  of  a  cylindrical  rinir. 

RULE. 

i     To  the  thickness  of  a  ring  add  the  inner  diameter 

II.  Multiply  this  sum  hy  the  square  of  half  the  thickness,  and 
\he  product  by  9.869G,  the  result  will  he  the  required  solidity. 

EXAMPLES. 

1.  What  is   the  solidity  of  an  anchor  ring,  whose  inner  di- 
ameler  is  8  inches,  and  thickness  in  metal  3  inches  ? 

8-f3  =  ll:    then,     11  x(^)^X 9.8696=244.2720,  which    ex 
presses  the  solidity  in  cubic  inches. 

2.  The  inner  diameter  of  a  cylindrical  ring  is  18  inches, 
and  the  thickness  4  inches  :  wl.at  is  the  solidity  of  the  ring  ? 

A'ls.  868.5248  aibic  inches 


262  APPLICATIONS 


Mensuration    of    Cylindrical    Rings. 

3.  Required  the  solidity  of  a  cylindrical  ring  whose  thick* 
ness  is  2  inches,  and  inner  diameter  12  inches  ? 

Ans.  138.1744  cubic  in 

4.  What  is  the  solidity  of  a  cylindrical  ring,  whose  thick- 
ness is  4  inches,  and  inner  diameter  16  inches? 

Ans.  789.568  cubic  m. 

6.  What  is  the  solidity  of  a  cylindrical  ring,  whose  thick- 
ness is  8  inches,  and  inner  diameter  20  inches  ? 

Ans.  

6.  What  is  the  solidity  jf  a  cylindrical  ring  whose  thick 
ness  is  5  inches,  and  inner  diameter  18  inches  ? 

Anfi,  


A    TABLE 

OF 

LOGARITHMS  OF  NUMBERS 
From  1  to  10,000 


K. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

I 

o-oooooo 

26 

I -414973 

5i 

\:]V^3 

76 

1-880814 

a 

o-3oio3o 

11 

1-431364 

52 

77 

I -886491 
1-892085 

3 

o-477'2« 

1-447158 

53 

1-724276 

78 

4 

0- 602060 

29 

i-4"2398 

54 

1-732394 

79 

1-897627 
1-903090 
1-908485 

5 

0- 698970 

3o 

i-477«2i 

55 

1-740363 

80 

6 

0-778151 

3i 

i-49'362 

56 

I -748188 

81 

I 

0-845098 

32 

i-5o5i5o 

57 

1-755875 

82 

i-9i38i4 

0-903090 

33 

i-5i85i4 

58 

1-763428 

83 

1-919078 

9 

0-954243 

34 

1^544^? 

59 

1-770852 

84 

1-924279 

10 

I • 000000 

35 

60 

1-778.51 

85 

1-929410 

II 

I -041393 
I -079181 
I-II3943 

36 

i-5563o3 

61 

1-785330 

86 

1-934498 

12 

ll 

I-568202 

62 

1-792392 

ti 

1-939519 

i3 

1-579784 

63 

I -799341 

1-944483 

14 

1-146128 

39 

i- 591065 

64 

1-806181 

89 

I .949390 

i5 

1-176091 

40 

I  -602060 

65 

1-812913 
1-819544 

90 

1.954243 

i6 

I -2041 20 

41 

1-612784 

66 

9' 

1-959041 
1-96.3788 

\l 

1.23O440 
1-255273 

1-278754 

42 

1-623249 
1-633468 

tl 

1-826075 

92 

43 

1-832309 

93 

1-968483 

'9 

44 

1-643453 

69 

1-838849 
1-84509S 
I -851258 

94 

1-973128 

20 

i-3oio3o 

45 

I -653213 

70 

95 

1-977724 

21 

1-322219 
1-342423 

46 

1-662758 

71 

96 

I -982271 

22 

ii 

1-672098 

72 

1-857333 

u 

1-986772 

23 

1-361728 

1-681241 

73 

1-863323 

1-991226 

24 

I -3802 II 

49 

1  -  690 1 96 

74 

1-869232 

99 

1-995635 

25 

1.397940 

5o 

1-698970 

75 

1-875061 

100 

2 ■ 000000 

Remark. — In  the  following  table,  in  the  nine  right- 
hand  columns  of  each  pnge,  where  the  first  or  lead- 
ing figures  change  from  9's  to  O's,  points  or  dots  are 
introduced  instead  of  the  O's,  to  catch  the  eye,  and  to 
indicate  that  from  thence  the  two  figures  of  the  Log- 
arithm to  be  taken  from  the  second  column,  stand  in 
the  next  line  below. 


2 

A  TABLE 

OF  LOGARITHMS  FROM  1 

TO 

10,000. 

N. 

0 

I 

2 

3 

4 

5 

6 

T 

8 

9 

1).  ' 

432 

100 

000000 

0434 

0868 

i3oi 

1734 

~2i66 

2598 

3o29 

3461 

3891 

lOI 

4321 

4751 

5i8i 

5609!  6o38 
98761  •3oo 
4100,  4521 

6466 

6894 

7321 

774S 

8174 

428 

102 

8600 

9026 
3259 

945 1 

368o 

•724 

1147 

1570 

1993 

24x5 

424 

io3 

012837 
7033 

4940 

536o 

5779 

6197 
•36 1 

6616 

419 

104 

745i 

7868 

8284 

8700 

9116 

3252 

9532 
3664 

9947 
4075 

4896 

416 

io5 

021 189 

i6o3 

2016 

2428 

2841 

4486 

413 

io6 

53o6 

5715 

6125 

6533 

6o42 

735o 

7757 

8164 

8571 

8978 

408 

IS 

o384 
033424 

VS^: 

•195 

•600 

1004 

1408 

1812I  2216 

2619 

302I 

404 

4227 

4628 

5029 

543o 

5830 

623o 

6629 

7028 

400 

?09 

7426 

78251  8223 

8620 

9017 

0414 
3362 

9811 
3755 

•207 

•602 

s 

396 

no 

ail  393 

1787  2182 

2576 

lltt 

4148 

4540 

3q3 

III 

5323 

5714 

6io5 

it 

ii53 

7664 

8o53 

8442 

112 

9218 
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A  TABLE 

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10,000. 

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4 

5  1  6  1  7 

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194 

224 

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6981;  7172'  7363;  7554 

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197 

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9076  9266 

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2671 

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4363 

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4739  4926 

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53oi 

188 

232 

5488 

5675 

6236 

6423 

6610  6796 
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6983 

7169 

187 

233 

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77291  7913 

8101 

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2  30 

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8701  8859'  9017 

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1 

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1 54 

282 

450249 

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283 

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2247  2400  .2553  2706'  2839 

3012;  3i65 

i53 

284 

33i8 

347. 

3624 

3777,  393c  4082  4235|  4387 

4540,  4692 

1 53 

285 

4H45 

4997 

5no 

5302'  5454  56o6  575s'  5910 

6062 

6214 

l52 

286 

6366 

65iS 

6670 

6S21  6973  7125  7276!  7428 
8336  8487'  B633  87S9  8940 

7579 

773. 

l52 

III 

7882 

80,33 

81S4 

9091 

9242 

i5i 

9392 

9543 

9694 

9S45  9995^  •146  •296  '447 

•597 

•748 

i5i 

389 

460898 

1048 

1198 

i348  1499  1649  1799'  1948 

2098 

224S 

i5o 

190 

462398 

2548 

26971 

2847  2997I  3i46  3296  3415 

l^i 

3744 

i5o 

2gi 

53?3 

4042 

'& 

4340  4490  4639  4788^  4936 

5234 

149 

29: 

5532 

5829  5977  6126  6274  6423 

6571 

6719 

'a 

293 

6868 

7016 

uti 

73121  7460  7608  7756  7904 
8790  8938  9085  9233  9380 

8o52 

8200 

294 

8347 

8495 
9960 
1438 

9527 

•998 

9675 

148 

295 

9822 

•116 

•263:  •410  •557'  •704  •85 1 

1 145 

147 

296 

'''.]ll 

1 585 

1732'  1878  2025  2171  23l8 

2464 

2610 

146 

!?J 

2903 
4362 

3o4o 
45o8 

3io5^  33411  3487  3633  3779 
4653  4799  4944  5090  5235 

3925 
538i 

4071 

146 

4216 

5526 

146 

299 

5671 

58i6 

5962 

6107  6232;  6397  6542  6687 
-»55d;  7700I  7844,  79^9'  81 33 
8999'  9143,  9287  94311  9575 
0433,  o582  0725  0869  1012 

6832 

6976 

145 

3oo 

477»2i 
8566 

7266 

74..' 

8278 

8422 

145 

3oi 

87.1 

8855 

97'9 

9863 

144 

302 

480007 
1443 

oi5i 

0294 

1156 

1299 

144 

3o3 

1 586 

1729 

1872'  2016  2159'  23o2  2445 

2588 

273. 

143 

3o4 

2874 

3oi6 

3.59 

3302!  3445,  3587  3730  3872 

4oi5 

4157 

143 

3o5 

43oo 

444'2 

4585 

4727;  4869  5oiil  5t53  5295 

5435? 

5579'  142 

3o6 

5721 

5863 

6oo5, 

6147  6289  643o  6572  6714 

6855 

6997,  >42 

12 

S551 

7280 

7421 

7563  7704  7845;  79S6  8127 

8269 

8410 

141 

8692 

8833 

8974  91 14'  9255  9396  9537 

9677 

98.8 

141 

309 

9958 

••99 

•239 

•38o  •J2o|  •66 r  •8oii  •g'.i 

1081 

1222 

140 

3io 

491362 

l502 

1642 

17S2:  1922  2062  2201:  234J 
3119'  3319I  3458.  3507  3737 
4572;  47"  485o!  49*59  5128 
5960;  6099!  6238;  6376'  65 1 5 
7344I  7483,  7621  7759'  7897 
8724!  8862  8999'  9137!  9275 

2481 

2621 

140 

3ii 

2760 

2900 

3o4o| 

3876 

401 5 

139 

3l2 

4i55 

4294 

4433; 

5267 

5406 

139 

3i3 

5544 

5683 

5822! 

6653 

6791 

\^ 

3i4 

tt. 

7068 

7206' 

8o35 

8173 

3i5 

8448 

8586| 

9412 

9550 

r38 

3i6 

9687 

9824 

9962 

••99j  ^236 

•374;  •5ii!  ^648 

•785 

•922 

.37 

3'7 
3id 

5oio59 

1 196 

1 333 

1470  1607 

1744  1880  2017 

2i54 

2291 
3655 

137 

2427 

2564 

2700 

2837  2073 
4199  4J35 

3109I  3246  3382 

35i8 

1 36 

3.9 

3791 

3927 

4o63, 

447 1 1  4607  4743 

4878 

5oi4 

1 36 

320 

5o5iDo 

5286 

542 1 1 

55D71  5693 

5828  5964  6099 
71B1  73i6  745i 

6234 

6370 

1 36 

3ai 

65o5 

6640 

6776, 

6911  7046 

7586 

7721 

i35 

322 

7856 

^11; 

8126 

8260  8395,  853o!  8664  8799 

8934 

9068 

i35 

323 

9^03 

9471 

9606  9740 

9874'  •••9  ^143 

•277 

•411 

1 34 

324 

5io545 

0679 

o3i3i 

0947  1081 

I2i5]  i349  1482 

1616 

1750 

1 34 

325 

1 883 

2017 

2l5li 

2284'  2418 

255i|  2684  281S 

2951 

3o84 

1 33 

3:5 

3218 

335i 

3484 

36171  3750 

3883  4016  4U9 

4282 

4414 

i33 

327 

328 

4548 

4681 

4813 

4946  5079 

5211  5344  5476 

5609 

3741 

i33 

5874 

6006 

6.39' 

6271  64o3 

6535  6668  6800 

6932 

7064 

l32 

319 

7196 

7328 

7460, 

7V'  7724 
8909  9040 

7855]  79871  8119 

825i 

8382 

l32 

33o 

5i85i4 

8646 

8777 

9171 

93o3.  9434 

9566 

9697 

i3i 

33i 

«9828 

9959 

••90' 

•221  •353 

•48' 

•6i5|  -745 

•876 

1007 

i3i 

332 

5jii38 

;^;'? 

1400 

i53o  1661 

1792 

1022  2d53 

2i83 

23i4 

i3i 

m 

2444 

2705; 

2835  2966 

3096I  32261  3356 

3486 

36i6 

i3o 

334 

3746 

3876 

4006 

4 1 36  4266 

4396 

4526!  4656 

4785 

4qi5,  i3o 

335 

5o45 

5i74 

5304* 

5434  5563 

5693 
6985 

5822:  5951 

6081 

D-^^io  !2g 

336 

6339 

64^9 

6=iQs; 

6727;  6856 

7114  7243 

8660 

75oi!  129 
8788  120 

337 

fi 

7739 

7SS3: 

8016  8145 

8274 

8402'  853 1 

338 

9045 

9' 74; 

9302  943o 

9559  96871  9815 

9943 

••72  ia8 
i3i)il  128 

339 

530200 

o328 

0456: 

o584  0712I  0840;  0968  1096 

1223 

8 

N. 

0 

1  I 

>  1 

3  i  4  i  5  1  6^   7 

9 

D. 

A    TABLE    OF    LOGARITHMS     FfloM     1    TO    10,000. 


K. 

0 

1 

2 

3 

4 

5 

6 

7  ! 

8 

9 

ur 

340 

531479 

1607 

1734 

1862 

1990 

2117 

2245 

2372: 

25oo 

2627 

128 

341 

2754 

2882 

3009 

3i36 

3264 

3391 

35i8 

3645 

3772 

3899 

127 

342 

4026 

4i53 

4280 

4407 

4534 

4661 

4-87 

4914 

5o4i 

5167 

127 

343 

5294 
6558 

5421 

5547 

5674 

58oo 

5927 

6o53 

6180 

63o6 

643a 

126 

344 

6685 

6811 

6937 

7063 

n 

73i5 

7441, 

7567 

& 

126 

345 

7819 

7945 

807. 

8197 

8322 

8574 

8699' 

8825 

126 

346 

9076 

9202 

9327 

9432 

9578 

9703 

9829 

9934 

••79 
i33o 

•2,14 

125 

347 

540329 

0455 

o58o 

0705 

o83o 

0955 

1080 

i2o5: 

14^^^ 

125 

34« 

1579 

1704 

1829 

1953 

20l8 

2203 

2327 

2452 

2576 

27.1 

125 

349 

282! 

2950 

3074 

3199 

3323 

3447 

3571 

4936; 

3820 

3944 

124 

O'JO 

544068 

4192 

43 16 

4440 

4564 

4688 

4812 

5o6o 

5i33 

124 

35 1 

5307 

5431 

5555 

5678 

58o2 

5925 

6049 

6172: 

6296 

6419 

124 

35; 

6543 

6666 

6789 

6913 

7o36 

7.59 

7282 

74o5, 
8635 

m 

7652 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

85i2 

8881 

123 

354 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861  i  9984 

•106 

123 

355 

550228 

o35i 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

122 

356 

i45o 

1572 

1694 

1816 

1938 

2060 

2l8l 

23o3, 

2425 

2547 

122 

iil 

2668 

2790 

2911 

3o33 

3i55 

3276 

3398 

35i9 

3640 

3762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

47311 

4H52 

4973 

121 

359 

5094 

52i5 

5336 

5457 

5578 

5699 

5820 

5940' 

6061 

6.82 

121 

36o 

5563o3 

6423 

6544 

6664 

6785 

6903 

7026 

8349' 

7267 
8469 

l?J, 

120 

361 

7507 

7627 
8829 

7748 

7868 

7988 

810S 

8228 

120 

36a 

8709 

8948 

9068 

9188 

93o8 

9428 

9548, 

9667 

9787 

120 

363 

9907 

••26 

•146 

•265 

•385 

•5o4 

•624 

•743 

•863 

•982 

119 

364 

56II01 

1221 

i34o 

1459 

1078 

1698 

18.7 

1936 

2o55 

2174 

119 

365 

2293 

2412 

253i 

2o3o 

2769 

2887 

3oo6 

3i25, 

3244 

3362 

119 

366 

3481 

36oo 

3718 

3837 

3933 

4074 

4192 

43ii 

4429 

4548 

',\t 

367 

4666 

4784 

4903 

502I 

5i39 

5257 

5J76 

5494! 

56i2 

5730 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673: 

6791 

6909 

118 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849' 

7967 

8084 

118 

370 

568202 

83i9 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

•17 

371 

9374 

9491 

9608 

9725 

9842 

9959 

••76 

•■9^: 

•309 

•426 

•17 

372 

570543 

0660 

0776 

0893 

1010 

1126 

1243 

1339' 

1476 

1% 

117 

373 

1709 

2872 

1825 

1942 

2o58 

2174 

2291 

2407 

25231 

2639 

116 

374 

2988 

3io4 

3220 

3336 

3432 

3568 

3684 1 

38oo 

3915 

116 

375 

4o3i 

4147 

4263 

4379 

4494 
565o 

4610 

4726 

4841 1 

4957 

5072 

116 

376 

5i8S 

53o3 

5419 

5534 

5765 

588o 

5996; 

6111 

6226 

ii5 

377 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

ii5 

378 

U% 

7607 

7722 

7836 

7951 

8066 

8i8i 

8295; 

8410 

8525 

1 15 

379 

8754 

8868 

8983 

9097 

9212 

9326 

9441' 

9555 

9669 

114 

3bo 

579784 

9898 
1039 

••12 

•126 

•241 

i355 

•469 

•583 

•697 

•81I 

114 

38i 

580925 

ii53 

1207 

i38i 

1493 

1608 

1722 

2§58 

1 836 

1950 

114 

382 

2063 

2,77 

2291 

2404 

25i8 

26ii 

2-45 

2972 

3o85 

114 

383 

^3^ 

33i2 

3426 

3539 

3652 

3765 

3879 

3992 

4io5 

4218 

ii3 

384 

4444 

4557 

4670 

4783 

4896 

5009 

5l22 

5235 

5348 

ii3 

385 

546! 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

n3 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374^ 
8496 

7486 
8608 

7599 

1 12 

387 

m'. 

7823 

7935 

8047 

8160 

8272 

8384 

8720 

1 12 

388 

8944 

90  56 

9167 

9279 

9391 

95o3 

9615 

9726 

9838 

1 12 

389 

9950 

••61 

•173 

•284 

•396 

•5o7 

•619 

•730 

•8/.2 

•953 

1 12 

390 

591065 

1176 

1287 

1399 

i5io 

1621 

1732 

1843 

,955 

2066 

I  n 

39» 

2177 

2288 

2399 

25lO 

2621 

2732 

2843 

2954 

3o64 

3.75 

1 1 1 

]g'i 

3286 

3397 

35o8 

36i8 

3729 

3840 

3950 

4061 

4171 

41S2 

1 11 

l9i 

4393 

45o3 

4614 

4724 

4834 

4945 

5o55 

5i65 

5276 

538ff 

no 

^1 

5496 

56o6 

3717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

no 

3c,5 

2695 

6707 

6817 

6927 

7037 

7146 

7256 

7366, 

8462 

7476 

7586 

no 

396 

78o5 
8900 

1% 

7914 

8024 

8i34 

8243 

8353 

8572 

8681 

no 

399 

9883 
600973 

9009 

•lOI 

1191 

9119 
•210 

12Q9 

9228 
•319 
140S 

nil 

l5i7 

9446 
•537 
1625 

9556  9665 
•646  ^755 
1734'  1843 

'.Hi 

1951 

109 
109 
109 

N. 

2  o_ 

1 

» 

3 

4  ^ 

^_ 

6 

7  ! 

8_ 

9 

D. 

A  TiBLE 

OF 

LOGARITHMS  FROM  ] 

TO 

10,000. 

It 

N. 
400 

0 

• 

"'~' 

3 
2386 

4  i  6 

2494   2603 

6 

-Tr»- 

9  |D. 

3o36  108 

602060 

2160  2277 
32d3  336i 

2711 

2819  2928 

401 

3i44 

346g 

3577  3686  3-194 

3902  4010 

4.18  .08 

402 

4226 

4334  4442 

455o 

4658  ijbb    4874 

4982!  5089 

5.97  .08 

4o3 

53o5 

54i3 '5521 

5628 

5136  5844  5931 
681 1  6919  7026 

6059  6166 

7i33:  724. 

6274'  «o8 

404 

638 1 

6489  6596 

6704 

7348,  .07 
84.9!  37 
9488,  107 

4o5 

7455 

7562  7669 
8633,  8740 

8847 

7884 
}^954 

799 1|  8098 

82o5j  83.2 

406 

8526 

906 II  9167 

9274,  938. 

407 

^  9594 

970 1  j  9808 

99' 4 

••21 

•128,  •234 

•34 1 1  •447 

•554'  107 
.617  106 

4o3 

610660 

0767,  0873 
1829  1936 

0979 

1086 

1192 

1298 

i4o5;  i5ii 

4f>9 

1723 

2042 

2148 

2254 

236o 

2466;  2572 

26781  106 

410 

612784 

2.S90  2996 
3947  40D3 

3l02 

3207 

33.3 

34.9 
4473 

35251  3630 

3736'  .06 

411 

3842 

ii^ 

4264 

4370 

458.!  4686 

47921  106 

412 

4897 

5oo3,  5 108 

5319 

5424 

5529 

5634 

5740 

5845!  io5 

4i3 

5960 

6od5|  6160 

6265 

6370 

6476 

658. 

6686 

s 

6895  Jo5 

414 
415 

7000 
8048 

7io5  7210 
8153!  6257 

73.5 
8362 

7420 
8466 

7525 
857. 

8676!  8780 

7943  io5 
8989  io5 

416 

9093 

9198;  9302 

9406 

95ii 

9615 

9719 

9H24 

9928 

••321  104 

^'1 
4i8 

620106 

0240!  o344 

0448 

o552 

o656'  0760 

0864 

0968 

.072 

104 

1 1 76 

1280  1 384 

1488 

1592 

.695 

»799 

.903 

2007 

2..0 

.04 

419 

2214 

23i8 

2421 

2525 

2628 

2732 

283d 

2939 
3973 

3o42 

3.46 

.04 

420 

623249 

3353 

3456 

3559 

3663 

3766 

3S69 

4076 

4179 

io3 

421 

4282 

4385 

4488 

4591 

4695 

479« 

4901 

5oo4 

5.07 
6.35 

52.0 

io3 

422 

53i2 

54i5 

55i8 

5621 

5724 

5827 

5929 

6o32 

62381  .o3  1 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

& 

7.6. 
8.85 

7263 

io3 

424 

7366 

7468 
8491 

7571 
8593 

mi 

^797 
9817 
0835 

7878 
8900 

7980 

8287 
9308 

102 

425 

8389 

9002 

9.041  9206 

102 

426 

94io 

95i2 

9613 

971 5 

9919 

••2. 

•.23 

•224 

•326 

.02 

^l 

630428 

o53o 

o63i 

0733 

0936 

.038 

;;^? 

124. 

.342 

102 

1444 

1 545 

1647 

1748 

1849 

.951 

2052 

2255 

2356 

10. 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3o64 

3.65 

3266 

3367 

101 

43o 

633468 

457? 
5584 

3670 

3771 

4880 

3973 

4074 

4.75 
5.82 

4276 

4376 

.00 

43 1 

5484 

4679 

i]it 

498. 

5o8i 

5283 

5383 

100 

432 

068D 

5886 

5986 

6087 

6.87 

6287 

6388 

100 

433 

6488 

6588 

6688 

6789 

6S89 

6989 

Co 
9088 

7189 

7290 
8290 
9287 

9387 

ICO 

434 
435 

1% 

9486 

l^ 

a° 

7790 
8780 
9785 

& 

7990 
89H8 

8.90 
9.88 

99 
99 

436 

9586 

9686 

9885 

9984 

••84 

•i83 

•283 

•382 

99 

43^ 

640481 

o58i 

0680 

0779 

0879 

0978 
.970 

1077 

"77 

1276 

.375 

99 

1474 

1573 

1672 

1771 

1871 

2069 
3o58 

2168 

2267 

2366 

99 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3.56 

3255 

3354 

^ 

440 

643453 

355i 

365o 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

44i 

4439 

4537 

4636 

4734 

4832 

4931 

5o29 

5.27 

5226 

5324 

98 

442 

5422 

5521 

56i9 

5717 
6698 

58j5 

59.3 
6894 

601 1 

6.10 

6208 

63o6 

98 

443 

6404 

65o2 

6600 

6796 

6992 

7089 
8067 
9043 

^!?5 

7285 

98 

444 

7383 

7481 
8458 

7579 

7676 

7774 

7872 
8848 

& 

8262 

98 

445 

8360 

8553 

8653 

8750 

9.40 

9237 

97 

446 

9335 

9432 

9530 

9627 

9724 

982. 

X2 

••16 

•..3 

•2.0 

97 

ni 

65o3o8 

o4o5 

o5o2 

0599 

0696 

0793 

0987 

1084 

1.8. 

97 

1278 

1375 

1473 

069 

1666 

.762 

18D9 

1956 

2o53 

2.5o 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

30.9 

3. .6 

^9^ 

430 

653213 

3309 
4273 

34o5 

3502 

3598 

3695 
4658 

379. 

3888 

3984 

4080 

45 1 

mi 

4369 

4465 

4562 

4734 

485o 

4946 

5o42 

96 

452 

5235 

533 1 

5427 

5523 

56.9 

57.5 

58io 

6?64 

6002 

96 

453 

6098 
7o56 

6194 

6290 

6386 

6482 

6577 
75i4 

6673 

6769 
7725 

6960 

96 

454 

7i52 

7247 

7343 

7438 

itr. 

7820 

K  $ 

455 

8011 

8107 

8202 

8298 
925o 

8393 

8488 

8679  8774 

456 

8965 

9060 

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9346 

944. 

9536 

963 1-  9726' 

9821   95 

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•201 

•296 

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•486 

•58.  •676 

•771   95 

0960 

io55 

n5o 

1245 

.339 

1434 

.529'  1623 

1718I  95 

459 

i8i3 

1907 

2002 

2096 

2191 

2286  238o 

2473;  2569' 

2663 j  95 

N. 

0 

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4  !  5  1  6 

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A    TABLB    OF    LOGAFITilMS     FROM    1    TO    10,000. 


ri^r 

0 

1    I 

1  2  j  3  1  4  1  5  1  6  1  7  1  8  I  9    1>.  | 

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662758 

7852 

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401 

3701 

3795 

:  38»9'  3783,  4078  4172 

4266  436o  4454:  454fe 

94 

462 

4642 

4736 

1  4830 

1  4324'  5oi8  5ii2 
1  5862 J  5906;  6o5o 

5206  5299  5393,  5487 

94 

463 

5581 

5675 

:   5769 

6143  6237  633i  6424 

94 

464 

65i8 

6612!  6703 

j  6799  6892  6986 
7733  7826  7920 

7079;  7 '73:  7266,  736a 

94 

465 

1453 
8386 

7346 

7640 

8oi3!  8106  8199  8293   93 

466 

8479 

8572 

8665.  8759  8852 

1  89451  9o38  9i3i  9224!  93 

467 
468 

9317 

9410 

95o3 

9596!  9689 

:  9782 

9875!  99671  ••60'  •i53t  93 

070246 

0339 

043 1 

o524!  0617 

0710 

1  0802 

0895;  0988:  io8oi  91  i 

469 

1173 

1265 

i358 

i45ii  i54'j 

i636:  1728 

1821 

1913 

2836 

2oo5  9^  1 

470 

672098 

2190 

2283 

2375,  2467 

256o 

2652 

2744 

2929  92 

4-1 

3o2i 

3ii3 

32o5 

3297J  3390 

3482 

3574 

3666 

3758 

38Jo,  92 

47  i 

3942 
4861 

4034 

4126 

42i8l  43io 

4402 

;  4494 

4586 

4677 

4769  92 

473 

4953 
5870 

5o45 

5i37  5228 

5320 

5412 

55o3 

5595 

5687   92 

474 

5778 
6694 

5962 
6876 
7789 
8700 

t)o53|  6145 

6236 

6328 

64«9 

65u 

6602   92 

475 

6785 

6968!  7059 

7i5i 

7242 

7333 

7424 

7516 
8427 

9' 

476 

2^"2 

7698 

7881 
8791 

8882 

8o63 

81 54 

8245i  8336 

91 

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478 

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8609 

8973 
9882 

9064 

9i55j  9246 

9337 

9' 

94?8 

9519 

9610 

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9973 

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91 

480 

68o336 

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0698 

0789 

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0970 

1060 

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91 

681241 

i332 

1422 

i5i3 

i6o3 

1693 

1784 

1874 

1964 

2o55 

90 

481 

2145 

2235 

2326 

2416 

25o6 

2596 

2686 

2777 

2867 

2957 

90 

482 

3o47 

3i37 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

3947 

4037 

4127 

4217 

43o7 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845 

4935 

5o25 

5ii4 

5204 

5294 
6189 
7o83 

5383 

5473 

55fe3 

5652 

?9 

485 

5742 

583 1 

?i 

6010 

6100 

6279 

6368 

6458 

6547 

486 

6636 

6726 

6904 

^t 

8064 

7261 
81 53 

7351 

7440 

89 

487 

7329 
8420 

7618 

6dq8 

7796 

m 

8242 

833 1 

89 

488 

85o9 

8687 

8776 

8953 

9042 

9i3i 

9220 

89 

489 

9309 

93n8 
0283 

94S6 

9575 

9664 

9753 

Q841 

9930 

••19 

•107 

^'^ 

490 

690106 
1081 

0373 

0462 

o55o 

0639 

0728 

0816 

0900 

-V093 

89 

491 

1 1 70 

1258 

i347 

1435 

i524 

1612 

1700 

•  780 

1877 

88 

492 

1965 

2o53 

2142 

223o 

23i8 

2406 

2494 

2583  2671 

2759 

88 

493 

1847 

2935 
38i5 

3o23 

3iii 

3'99 

3287 
4166 

3375 

3463  355i 

3639 

83 

494 

3727 

3903 

3991 

4078 

4254 

4342  443o 

4517 

88 

495 

4605 

4693 

4781 

4868 

4956 
5832 

5o44 

5i3i 

5219  5307 

5394 

88 

496 

5482 

5569 

5657 

5744 

5919 

6007 

6004!  6182 

6269 

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497 

6356 

6444 

653 1 

6618 

6706 

6793 

6880 

6968!  7055 

7142 
8014 

l^ 

498 

7229 

tsl 

7404 

7491 

7578 
8449 

7665 

7752 

7839  7926 
8709  8796 

t^ 

499 

8101 

8275 

8362 

8535 

8622 

8883 

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698970 

9057 

9144 

923. 

9317 

9404 

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9578  9664 

975i 

87 

5oi 

9838 

9924 

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••98  •184 

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87 

502 

700704 

0790 

0877 

0963  io5o 

ii36 

1222 

1309I  13^5 

2172I  2238 

1482 

86 

5o3 

1 568 

i6d4 

1741 

1827  1913 

1999 

2086 

2344 

86 

5o4 

243 1 

25i7 

26o3 

2689!  2775 

2861 

2917 

3o33  3 1 19 

32o5 

86 

5o5 

3291 

3377 

3463 

3549!  3635 

3721 

3807 

3893 

3979 

4o65 

86 

5o6 

4i5i 

4236 

4322 

44o8i  4494 

4579 

4665 

4731 

4837 

4922   86  1 

507 

5008 

5094 

5179 

5265|  5350 

5436 

5522 

56o7 

5693 

5778 

86 

5o8 

5864 

'r, 

6o3d 

6120  6206 

6291 

6376 

6462,  6547! 

6632 

85  1 

509 

6718 

6888 

6074  7059 

7144 

7229 

73 1 5  7400I 

7485 

85 

5io 

707570 

7655 

7740 

7996 

8081 

8166  825i 

8336 

85 

5ii 

8421 

85o6 

8591 

8846 

8931 

90 1 5  9100; 

9185 

85  , 

5ia 

9270 

9355 

9440 

9524'  9609 

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9863  9948| 

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710117 

0202 

0287 

037 1 j  0456 

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0710  0794: 

0879 

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1048  Il32 

1217  i3oi 

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1723   84  1 
2566   84  1 

5i5 

1892 
2734 

1976 

2060'  2144 

2229  23i3 

2397  2481 1 

5i6 

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2§.8 

2902'  2986 

3070:  3i54 

3238 

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3407 

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3575;  3659 

3742;  3826 

3910  3994 
4749!  4833 

4078 

4162 

4246 

84 

44i4i  4407  458 1  4665 
525i|  5335  5418  55o2 

4916  5ooo 

5oS4 

84 

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5753!  5836 

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84 
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I  I  2  !  3  1  4  ;5  i 

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7  i  8  ! 

9 

TAIiLK    OF    LOGAIUTI1M8    FROM    1    TO     10,000. 


~N.* 

0   1  I  1  2 

3 

4 

5 
6421 

6" 

7 

8  I  , 

D.  1 

520 

716003  0087  6170 

6254 

6337 

6304 

6588 

6671!  6754  S3 
7504!  7^87!  «3 

521 

6838,  6921  7004 

7088 

Ti7il  7254  7338 
800 3  8086'  8169 
8834  891 71  9000 

7421 

523 

523 

83o2 

mjm  1% 

8253  83361  8419:  83 
9o83]  9i65j  9248,  83 

524 

933. 

9414'  9497 

9580 

9663,  9745i  9S28 

99111  9994!  ••77,  83 
0738:  082 i|  0903I  83 
1563  164&'  1728,  h 

525 

720159 

0242,  0323 

0407 
1233 

0490'  0373!  o655 

'J26 

0986 

1068 

n5i 

i3i6:  1398 

1481 

527 

iBi. 

1893 

.975 

2o58 

2i4o|  2222 

23o5 

2387 

2469 

2552   d?  i 

52a 

2634 

2716 

2> 

2881 

29O3,  3o45 

3127 

3948 

3209 

329. 

3374 

82 

529 

3456 

3538 

3620 

3702 

3i84'  3866 

4o3o 

4112 

4194 

8j 

53o 

724276 

4358 

4440 

4322 

4604!  4685 

m 

4849 

4931 

5oi3 

82 

53 1 

5095 

5176 

5258 

5340 

5422'  5303 

56^7 
6483 

5748 
6564 

583o 

82 

532 

5912 

5993 

6075 

61 56 

6238,  632o'  6401 

6646 

82 

533 

6727 

6809 

6890 

6972 

7033  7134  7216 
7866  7948  8029 
8678:  8739  8841 

7297 

8110 
8922 

7379 
819. 
9003 

7460 
8273 
9084 

81 

534 

5iD 

7041 
8354 

7623 
6435 

7704 
85i6 

nil 

81 
8i 

536 

9165 

9246 

9327 
•i36 

9408 

9480  9370 
.2981  ^378 

IIOD,  1186 

9651 

9732 

9813 

9893 

81 

537 

9974 

••55 

•217 

1266 

•340 

•621 

•702 
i5oS 

81 

538 

73o«2 
i589 

oS63 

0944 

1024 

i347 

1428 

81 

539 

1669 

1750 

1 830 

1911 

1991 

2072 

2l52 

2233 

23.3 

81 

540 

732394 

2474!  2555 

2635 

2715 

llfs 

2876 

2956 

3o37 

3117 

80 

541 

3197 

3278;  3358 

3438 

35.8 

3679 

3759 
4560 

3839 

3919 

80 

542 

3999 

4079,  4 1 60 

4240 

4320 

4400 

4480 

4640 

4720 
55.9 

80 

D43 

4800 

48S0!  4960 

5o4o 

5.20 

5200 

5279 

5359 

5439 

80 

544 

5599 

5679;  5739 

5838 

5918 

5998 

6078 

6i57 

6237 

63i7 

80 

545 

6397 

6476 

6556 

6635 

67.5 

73.1 

63o5 

8384 

6874 

6954 

7o34 

71.3 

80 

545 

7193 

6067 

7352 
6146 

7431 

S3 

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7908 

79 

54-' 

9672 

6225 

8543 

8701 

79 

548 

8860 

8939 

9018 

90971  9177 

9256 

9335 

9414 

nil 

79 

549 

965 1 

9731 

9810 

9889  9968 

••47 

•126 

•2o5 

79 

55o 

740363 

0442 

0321 

0600 

0678 

0757 

o836 

09.5 

0994 

1782 

2568 

1073    79  1 

55 1 

Il52 

I230 

288a 

1 388 

.467 

1546 

1624 

1703 

i860 

79 

552 

1939 

2018 

2175 

2254 

2332 

2411 

2489 

2647 

l^ 

553 

2723 

2804 

2961 

3o3o 
3823 

3n8 

3196 
39S0 

3270 

3353 

343. 

554 

35io 

3588 

3667 

3745 

3902 
4684 

4o58 

4.36 

42i5 

78 

555 

4293 

4371 

4449 

4328 

4606 

4762 

4840 

4919 

4997 

78 

556 

5075 

5.5] 

5231 

5309 

5387 

5465 

5543 

5621 

5599 

im 

7« 

557 

5855 

59331  601 1 

6089 

6167 

6245 

6323 

6401 

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6634 

6712;  6790 

6S68 

6945 

7023 

710. 

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8o33 
8808 

7334 
8.10 

8885 

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559 
56o 

741a 
748188 

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m 

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7800 
8376 

gl? 

78 
77 

56 1 

8963 

9040 

91  !8 

9193 

9272 

935o 

9427 

9304 

9582 

9639 

77 

562 

9736 

9'i.4 

989I 

9968 

••45 

•123 

•200 

•277 

•354 

•43 1 

77 

563 

7=.o5o8 

o586 

0663 

0740 

i5io 

0SI7 

0894 

0971 

1048 

1.23 

1202 

77 

564 

1279 

1 356 

1433 

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1664!  1741 

1818 

1895 

2663 

1972 

77 

565 

2048 

2125 

2202 

2279 

2356 

2433 

25o9 

2386 

l^ 

77 

566 

2816 

2893 

2970 

3047 

3.23 

3  200 

3277 

3353 

3430 

77 

567 

3583 

366o 

3736 
45oi 

38i3 

3889 

3966 

4042 

4119 

4883 

4195 

4272 
5o36 

77 

568 

4348 

4425 

4578 

4654 

4730 

4807 

4960 

76 

569 

5lI2 

5189!  5265 

5341 

5417 

5494 
6256 

5570 

5646 

5722 

5799 

76 

5/0 

755875 

5q5i  6027 

6io3 

6180 

6332 

6408 

6484 

6560 

76 

57' 

6636^  671 2I  6788 

6864 

6940 

7016 

70.72 

7168 

7244 

8oo3 
8761 

7320 

8079 
8836 

76 

572 
573 

'it, 

m  iz 

7624 
8382 

7700 
8458 

^533!  leJ) 

Zi 

76 
-6 

574 

8912 
9668 

8988 1  9o63 

9139 

9214 

9290I  9366 

9441 

9517 

9392 

7$ 

575 

9743;  9819 
0498J  0573 

9894 

9970!  ••4:11  "ni 

G?5o 

•272 

•347 

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575 

76.1422 

0649 

0724'  0799!  0875 
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22:8:  23o3!  2378 

1025 

iroi 

^^ 

P,l 

1176 

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1402 

1702 

1778,  1853 

75 

'^t 

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2153 

2453 

25291  2604'  75 
3278'  3353i  75 

579 

2754  2829 

2904 

29781  3o53!  3128 

32o3 

,   N. 

0 

I    2 

3 

4  J  5  i  6 

7 

8  1  9_1  D.J 

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A   TABLE    OP    LOGARITHMS    FROM    1    TO    10,000. 


N. 

0 

3 

3 

4 

5 

6  1  7  1 

8 

9 

I), 

58o 

763428 

35o3 

"35^ 

3653 

3727 

38o2 

3877i  3952! 

4027 

4101 

58i 

4176 

425i 

4326 

4400 

4475 

455o;  4624I  4699' 
5296  5870'  5443 

4774 

4848 

582 

641 3 

4998 

5072 

5U7 

5221 

5520 

5594 

583 

5743 

58i8 

6636 

5966 

6041  6ii5|  6190 

6264 

6888 

584 

6487 

6562 

6710 

6785  6859I  698 J' 

7007 

7082 

585 

7i56 

7280 

7304 

7379 

7453 

7527  7601 1  7675, 

7749 

7823 

586 

7898 

7972 

8046 

8120 

%t 

8268:  8342J  8416! 

8490 

8564 

587 
588 

8638 

8712 

8786 

8860 

9008I  9082  91 56 
9746]  98201  9894; 
0484  o557|  068 1 j 
1220  1298'  1867 

9280 

9808 

I'i 

9377 
770115 

945 1 

9525;  9599 

96,3 

9968 

••42 

589 

0189!  0263 

o336 

0410 

0703 

0778 

5oo 

770852 

09261  0999 

1073 

1146 

1440 

i5i4 

591 

1087 

1661  1784 

1808 

1881 

1955  2028 

2102; 

2175 

2248 

592 

2822 

2895  2468 

2542 

26i5 

2688J  2762 

2835: 

2908 

2n8l 

5^3 

3o55 

8128  3201 

3274 

3848 

8421 

3494 

3567! 

8640 

37.3 

594 

3786 

886oi  8988 

4006 

4079 

4i52 

4225 

4298I 

4371 

4444 

595 

4517 

4390 

4663 

4786 

4809 
5588 

4882 

49551  5028; 

5,00 

5,73 

596 

5246 

58.9 

5392 

5465 

56io 

5688  5736' 

5829 

5902 

597 

5974 

6047 

6120 

6193 

6265 

6338 

641.  6488 

6556 

6629 

5^8 

6701 

6774 

6846 

6919 

6992 

7064 

7187  7209' 

7282 

7354 

599 

7427 

7499 

7572 

7644 
8368 

7717 

7789I  7862I  7984' 

8006 

8079 

600 

778151 

8224 

8296 

8441 

85i3 

8585  8658, 

8780 

8802 

601 

8874 

8947 

9019 

9091 

9168 

9286 

9808  9380; 

9452 

9324 

602 

9596 

9669 

9741 

9813 

9885 

9957 

••29!  •lOlj 

•.78 

•245 

6o3 

780817 

0889 

0461 

o538 

o6o5 

0677 

0749!  0821 
1468J  .540' 

0893 

0965 

604 

1087 

1109 

1181 

1258 

1824 

1896 

.6.2 

1684 

6o5 

n55 

.827 

1899 

1971 
2688 

2042 

2114 

2186 

2258: 

2829 

2401 

606 

2473 
8189 

2544 

2616 

2759 
8473 

2881 

2902 

2974' 

8046 

3.17 

607 

8260 

8882 

3408 

3546 

86.8 

36So 
440.3! 

876. 

8882 

608 

8904 

8975 

4046 

4118 

4189 

4261 

4332 

4475 

4546 

609 

4617 

4689 

4760 

4881 

4902 

4974 

5o45 

5. .61 

5.87 

5259 

610 

785830 

5401 

5472 

5548 

56.5 

5686 

5757 

582S1 

5899 

5970 

611 

6041 

6112 

6i83 

6254 

6825 

6896 

6467 

6538: 

6609 

6680 

612 

6751 

6822 

6893 

6964 

7o35 

7106 

7'77 

7248| 

73,9 

7890 

6i3 

7460 

758i 

7602 

7678 

7744 

7815 

7885 

7956 

8027 

8098 

614 

8168 

8289 

8810 

838 1 

8451 

8522 

8598 

86631 

8734 

8804 

6i5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9869 

9440 

95.0 

616 

958 1 

965 1 

9722 

9792 

9868 

9933 

•••4 

••74' 

•144 

•2.5 

70 

617 

790285 

08  56 

0426 

0496 

o567 

0687 

0707 

0778: 

0848 

0918 

70 

618 

0988 

1059 

1 1 29 

1199 

1269 

1840 

.4.0 

1480 

i55o 

1620 

70 

619 

1691 

1761 

i83i 

1901 

1971 

2041 

2111 

2.8l| 

2252 

2822 

70 

620 

792892 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

8092 

8162 

8281 

83oi 

8871 

8441 

85.1 

858i| 

865i 

372. 

70 

622 

3790 

386o 

8980 

4000 

4070 

4. 39'  4209 

4279 

4849 
5045 

44.8 

70 

623 

4488 

4558 

4627 

4697 

4767 

4836  4906 

4976 

5ii5 

70 

624 

5i85 

5254 

5324 

58o3 
6088 

5463 

5532I  5602 

56721 

5741 

58ii 

^ 

625 

5880 

5949 

6019 

6i58 

6227  ^297 

6366; 

6436 

65o5 

626 

6574 

6644 

6713 

6782 

6852 

6921  6990 
7614  7688 
83o5  8874 

7060' 

7.29 

7198 

69 

627 
628 

7268 

7337 

7406 

7475 

7545 

7752; 

7821 

7800 
8582 

69 

7960 

8029 

8098 
87^0 
9478 

8167 
8858 

8286 

8443, 

85.3 

69 

629 

865 1 

8720 

8927 

8996 1  9065 

9134' 

9208 

9272 

^ 

63o 

799341 

9400 
0098 

9547 

9616 

9685!  9754 

9828: 

9892 

9961 

69 

63i 

fi 30029 

0167 

0236 

o8o5 

0873  0442 

o5 1 1 1 

o58o 

0648 

69 

63a 

0717 

0786 

o854 

0923 

0992 

1061 

1129 
i8i5 

1 198! 

1884^ 

1266 

1335 

69 

633 

1404 

1472 

i54i 

1609 

1678 

1747 

.952 

2021 

69 

634 

2089 

2i58 

2226 

2295 

2363 

2482 

25oo 

2568, 

2687 

2705 

^ 

635 

2774 

2842 

2910 

2979 

3o47 

3.16'  8184 

3252, 

3321 

3389 

636 

3457 

3525 

3594 

3562 

8780 

8798,  8867 

8935 

4008 

4071 

68 

637 
638 

4189 

4208 

4276 
4957 

4344 

4412 

4480  4548  4616 

4685 

4753 

68 

4821 

4889 

5o25 

5093 

5i6ij  5229  5297 

5365 

5433 

68 

639 

35oi 

556^ 

5637 

5705 

5773 

5841 1  5908  5976 

6044 

611J 

^\ 

N. 

■  - 
0 

1 

^^ 

3 

4 

5  1  6    7  1 

8 

^1— 

1). 

A    TABLE    OP    LOOAKITHSdG    FROM    1    TO    10,000. 


n 


N. 

0   1  ,  j  2 

3 

4    5    6 

7  1  8  1  9 

D.  j 

640   806180'  6348  63i6 

6384 

645i  65i9  6587 

66551  6723  6790 

68  1 

641 

6858 

6926!  6994 

7061 

7129  7«97  7264 

7332  7400 

I^^J 

68 

642 

7535 
8211 

7603 1  jt>io 
8279!  8346 

7738 
8414 

7806  7873  794, 
848,  85491  86,6 

8008  8076 

8,43   68 

643 

8684  8751 

88,8  67 

644 

88S6 

8953  9021 

9088 

9i56  92231  9290 

9358;  9425 

9492  67 

643 

9360 

9627 1  9694 

9762 

&i\  tt\  t^i 

••3,1  ••98 

•,65 

67 

646 

810233 

o3ooi  o367 

0434 

0703 1  0770 

fdl 

J*^ 

647 

0904- 

0971 j  io39 

1106 

ii-ji    ,240  ,3o7 

1374!  1441 

67 

648 

1575 

1642  1709 

1776 

1843  19101  ,977  2044^  21,1 

2178 

67 

649 

2245 

23l2  2379 

2445 

25,2  9379  2646'  2713 

2780 

2847 

^7 

6DO 

812913 

2980  3o47 

3114 

3i8,  3247  33i4 

338, 

3448 

35i4 

67 

65i 

3d8i 

3648 

3714 

3781 

3848  3914  3981 

4048 

4114 

4181 

67 

65i 

4248 

43i4 

438i 

4447 

43,41  438, I  4647 

4714 

4780 

4847 

67 

653 

4913 

4980 

5046 

5m3 

f/79!  5246:  53,2 

5378 

5445 

55,, 

66 

654 

5578 

5644 

5711 

5777 

58431  59,0!  5976 

6042 

6,09 

6,75 

66 

655 

6241 

63o8 

6374 

644o 

65o6l  6373 1  6639 

6705 

677, 

6838 

66 

656 

6904 

6970 

7o36 

7102 

7169!  7235i  73o, 

l^^l 

7433 

8820 

66 

tu 

7365 
8226 

7631 

^3?8 

7764 
8424 

7830 
8490 

m 

7962 
8622 

8028  8094 
8688  8734 

66 
66 

659 

8885 

9017 

9083 

9,49 

"^vi 

9281 

9346 

9412 

'^ll 

66 

660 

819344 

9610 

9676 

;S 

9807 

9873 

9939 

•••4 

••70 

66 

661 

820201 

0267 

0333 

0464 

033o 

03^3 

066, 

0727 

0792 

66 

662 

o858 

0924 
1D79 

0989 

,,20 

,,86 

123, 

i3i7 

,382 

1448 

66 

663 

i5i4 

i64D 

1710 

1775 

,84. 

1906 

1972 

2037 

2io3 

65 

664 

2168 

2233 

2299 

2QD2 

2364 

243o 

2493 

2  560 

2626 

2691 

2756 

65 

665 

2822 

2H87 

3oi8 

3o83 

3,48 

32,3 

3279 

3344 

3409 

65 

666 

3474 

3539 

36o5 

1  3670 

3735 

38oo 

3865 

3930 

3996 

406, 

65 

^ 

4126 

4i9« 

4256 

432, 

4386 

445, 

45,6 

458, 

4646 

471, 

65 

4776 

4B41 

nt 

4971 

5o36 

5,0, 

5,66|  523, 

5296 

536, 

65 

669 

5426 

5491 

562, 

5686 

6^9^ 

58,5  5880 

5945 

6o,o 

65 

670 

826075 

6140 

6204 

6269 

6334 

6464 i  6528 

6393 

6658 

65 

671 

6723 

6787 

6852 

6917 

698, 

7046 

7,,, 

1  7«73 

7240 

73o5 

65 

672 

7434 

7499 

!  7563 

7628 

1^^ 

7757 

7821 

7886 
853, 

795, 

65 

673 

8080 

8144 

'  8200 
'  8853 

8273 

8338 

84021  8467 

8395 

64 

674 

8660 

8724 

8789 

X 

8982 

9046  911, 

9,75 

9239 

64 

67D 

9304 

9368 

9432 

1  9497 

9625 

9600  9754 

9818 

9882 

64 

676 

9947 

**i  I 

••75 

:  •,39 

•204 

•268 

•3321  •3q6|  •460 

•525 

64 

678 

830D89 

0653 

0717 

1  0781 

0845 

0909 

09731  IO37J  ,,02 

1,66 

64 

I230 

1294 
1934 

i358 

1  1422 

,486 

,330 

,6,41  16781  ,742 

,8o6j  64 

679 

1870 

1998 
2637 

2062 

2,26 

2,80 
2828 

2253!  23i7  238, 

2445|  64 

680 

832509 

2673 

j  2700 

2764 

2892I  2956  3020 

3o83 

64 

681 

3i47 

3211 

3275 

1  3338 

3402 

3466 

353ol  35931  3657 
4,66'  423o  4294 

372, 

64 

682 

3784 

3848 

3912 

1  3975 

4o39 

4io3 

'  4357 

64 

683 

4421 

4484 

A-US 

1  461, 

4673^  4739 
53, o|  5373 

4802  4866  4929 
5437  55oo!  5564 

4993 

64 

684 

5o56 

5l20 

5i83 

i  5247 

5627 

63 

685 

6691 

5754 

58i7 

588  i 

5944I  6007 

6071I  6,34'  6,97 
6704!  6767 j  6830 

,  6261 

63 

680 

6324 

6387 

6451 

1  65,4 

63771  6641 
7210  7273 

'  68941  63 

687 
688 

tA 

7020 

7o83 

7146 

7336  7399  7462 

1  7525I  63 
8,561  63 

7652 

l]\l 

7778;  7841  7904 
8408  8471  8334 

7967  8o3o,  8093 
8597  8660  8723 

689 

8219 

8282 

8786;  63 

691 

638840 
9478 

8912 
9341 

8975 
9604 

1  9o38j  910,  9164 

;  9667,  9729'  9792 

9227  9280  o352 
9855  9918  9981 

94i5i  63 
••431  63 

692 

840106 

0169  0232 

I  0294  0357   0420 

0482  o;)45i  0608  06711  63 

693 

0733 

0796  0850 
1422  1485 

'  092,   0984   1046 

,,09  ,172!  ,234  13971  63 
1735,  1797  i860  ,0221  63 
236o  2422 1  2484  2347 i  62 

694 

\lsl 

;  1347 

16,0:  1672 

695 

2047!  2110 

2,72 

2235  2297 

696 

260c  26721  2734 
3233  3295I  3357 

2796 

2859  292, 

2o83  3046  3,08  3,70!  62 

^ 

3420 

3482:  3544 

36o6  3660!  3731!  37931  62 

3855!  3918:  3g8G 

4042 

4104  4166 

4229  42911  4353  441 5  6a  ! 

699 

N. 

44771  4539 

4601 

1  4664 

4726  4788',  485o  4912'  4974  5o36|  62 

0 

. 

3 

1  3 

1  4 

!  5 

1  6 

!  7 

1  8 

1  9 

1  r).  ) 

12 


A   TABLE    OF    LOGARITHMS    FROM    1     TO    10,00U. 


N. 

0 

I 

2 

3 

4 

5 

6 
5470 

7 

8 

9 

D. 

62 

7GO 

845098 

5i6o 

5222 

5284 

5346 

5408 

5532 

5394 

5656 

701 

5718 

5780 

5842 

5904 

5966 

602S 

6090 

6.5. 

62.3 

6273 
6894 

62 

-!02 

6337 

6399 

6461 

6523 

6385 

6646 

6708 

6770 

6832 

62 

703 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

75.1 

62 

704 

7573 

7634 

7696 

7758 

7819 

7881 
8497 

ifi> 

8oo4 

8066;  8128 

U 

1  7o5 

8189 

8231 

83i2 

8374 

843! 

8620 

86821  8743 

61 

706 

88o5 

8866 

8928 
9542 

8989 

905 1 

9112 

9'74 

9235 

9297 

9358 

61 

^S 

o-94'9 

9481 

9604 

9665 

9726 
o34o 

9788 

9849 

99" 

9972 

61 

830033 

0095 

oi56 

0217 

0279 

0401 

0462 

0324  o585 

61 

709 

0646 

0707 

0769 

o83o 

089; 

0932 

.014 

1075 

ii36 

"97 

61 

710 

851258 

l320 

i38i 

1442 

i5o3 

1 564 

1625 

1686 

'747 

.809 

6i 

711 

1870 

1931 

1992 

2o53 

2.14 

2175 

2236 

2297 

2353 

2419 

61 

712 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

713 

3090 

3i5o 

321. 

3272 

3333 

3394 

3455 

35.6 

3577 

3637 

61 

714 

3698 

3759 

3820 

388i 

3941 

4002 

4063 

4124 

4 1 85 

4245 

61 

715 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

716 

4913 

4974 

5o34 

5095 

5i56 

5216 

5277 

5337 

5398 

5459 

61 

7n 

5319 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6oo3 

6064 

61 

718 

6124 

6i85 

6245 

63o6 

6366 

6427 

6487 

6548 

6608 

6668 

60 

7>9 

6729 

6789 

685o 

6910 

6970 

7574 

703 1 

7091 

7i52 

7212 

7272 

60 

720 

857332 

7393 

7453 

75i3 

7634 

7694 

7755 

78.5 
8417 

7875 

60 

721 

7935 

7993 

8o56 

8116 

8176 

8236 

8297 

8357 

8477 

60 

722 

^Hl 

8397 

8657 

8718 

8778 

8838 

8898 

8958 

90.8 

9078 

60 

723 

9 1 38 

9198 

9258 

9318 

9370 

9i39 

9499 

9559 

9619 

9679 

•278 

60 

724 

0.9739 

0398 

9859 
0458 

99,8 

9978 

••38 

••98 

•i58 

•218 

60 

723 

86o338 

03l8 

0578 

0637 

0697 

0757 

08.7 

0817 

60 

726 

0937 

0996 

io56 

1II6 

1176 

1236 

1293 

i355 

I4i5 

1475 

60 

727 

1 534 

1394 

1 654 

I7I4 

1773 

i8'33 

1893 
2489 

1932 

2012 

2072 

60 

728 

2l3l 

2191 

225l 

23lO 

2370 

243o 

2549 

2608 

2668,  60 

729 

2728 

2787 

2847 

2906 

2966 

3o25 

3o85 

3.44 

3204 

3263i  60 

730 

803323 

3382 

3442 

3301 

356i 

3620 

3680 

3739 

3799 

3858 

59 

73  I 

3917 

3977 
4570 

4o36 

4096 

4i55 

4214 

42-:4 

4333 

4392 

4452 

59 

732 

45ii 

463  0 

4689 

4748 

4808 

4867 

4926 

49S5 

5o45 

59 

733 

5io4 

5i63 

5222 

5282 

5341 

5400 

5459 

56:9 

5578 

5637 

59 

734 

5696 
6287 

5755 

58i4 

5874 

5933 
6524 

5992 

6o5i 

6110 

6169 

6228 

59 

735 

6346 

64o5 

6465 

6383 

6642 

6701 

6760 

6819 

59 

736 

6878 

6937 

'^.tt 

7o55 

7114 

7173 

7232 

729. 

7350 

7409 

7998 

59 

737 

7467 

7526 
8ii5 

7644 

7703 
8292 

7762 
835o 

7821 

7880 

7939 

59 

738 

8o56 

B174 

8233 

8409 

8468 

8327 

8586 

59 

739 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9036 

9114 

9173 

59 

740 

869232 

9290 

9349 

9408 

9466 

9525 

95«4 

9642 

9701 

9760 

59 

741 

9818 

9877 

9935 

9994 

••53 

•ill 

•170 

•228 

•287 

•345 

742 

870404 

0462 

052l 

0570 

0638 

0696 

0755 

08.3 

0872 

oq3o 

58 

743 

0989 

1047 

1 106 

1 164 

1223 

1281 

1339 

IQ23 

1398 

I4561  13.5 

58 

744 

1673 

i63i 

l6qo 

1748 

1806 

1 865 

1981 
2564 

2040 

2098 
2681 

58 

745 

2 1 56 

22l5 

2273 

2331 

2389 

2448  25o6 

2622 

58 

746 

2739 

2797 

2855 

2913 

2972 

3o3o'  3o88 

3146 

3204 

3262 

58 

747 

3321 

3379 

3437 

3495 

3553 

36ii'  3669 

2727 

3785i  3844 

58 

748 

3902 

3960 

4018 

4076 

4i34 

4192'  4250 

43o8 

4366,  4424:  58 

749 

4482 

4540 

4398 

4656 

4714 

4772'  4830 

4888 

4945 

5oo3 

58 

750 

875061 

5i  19 
5698 

5,77 

5235 

5293 

535 1 1  5409 

5466 

5324 

5582 

58 

75i 

5640 

5756 

58i3 

5871 

5929'  5987 
65o7  6564 

6045 

6102 

6160 

58 

702 

6218 

6276 

6333 

6391 

6449 

6622 

6680 

6737 

58 

7-53 

6795 

6853 

6910 

6968 

7026 

7083,  7141 

7199 

7256 

7314 

58 

754 

7371 

7429 

7487 

7544 

7602 

7659'  7717 

7774 
8349 

7832I  7889 

58 

755 

7947 

8322 

8004 

8062 

8119 

8177 

8234I  8292 

8407 

8464 

57 

756 

8579 

8637 

8694 

8752 

8809  8866 
9383  9440 

8924 

8981 

9039 

57 

7Dd 

9096 

9153 

9211 

926S 

9325 

9497 

9555 

9612!  57 

9669 

9726 

9734 

9841 

9898 

9936  ••i3 

••70 

•127 

•i85 

57 

759 

N. 

880242 

0299 

0356 

o4i3 

0471 

o528'  o585 

0642 

0699 

0756 
9 

57 
D. 

0 

I 

l_i_. 

3 

4 

A±L 

7 

8 

A    lAULK    UF    LOGARITHMS    FROM    1    TO     10,000 


ib 


N. 

760 
761 
76a 

763 
764 
765 
766 

768 
769 
77c 
77« 
772 
773 
774 
775 
776 

77^  I 

779 

78c 

781 

782 

783 

7«4 

785 

786 

7«7 
788 
789 
790 

79" 
792 
793 

794 
795 
796 

797 
798 

799 
800 
801 
803 
8o3 
804 
8o5 
806 
807 
808 
809 
810 
811 
I  812 
8i3  I 
8.4 

8i:i 
81S 

8n  I 

8id  I 
819  I 

"n.  i 


880814 

i3S5 
1953 

2533 

3093 
366 1 
4229 
4793 
536 1 
5926 

886491 
7034 
7617 
8179 
8741 
9302 
9862 

89042 1 
0980 
!d37 

892095 
265 1 
3207 


087 1  j 

.442; 

2012 
23Si 

3i5o' 
3718 
4285 
4852i 
5418 
5983! 

6347' 
7iir, 

7674 
8236 

99i8j 

04771 
.o33^ 
.593' 

2  .do' 

2707 

„.„,,  3262, 

3762!  38.7, 
43i6j  4371 
4870  4923 
5423I  5478 
5975J  6o3o' 
6526;  658.^ 
7i32 
76H2 


0928'  0985 
1499!  i556 
2069!  2.26 
2638J  2695 
3207;  3264 
3775  3832 
4342I  4399 
49091  4960 
5474  5d3i 
6039  6096 
6604!  6660 
7.67;  7223 
773o'  7786 
8292;  8348 
88531  8909 


941. 
9974 


9470 
••3o 


7077 
S97627 
8.76 
8-125 
9273 
9821 
900367 
09.3 
.458 

2003 

2547 

903090 

3633 


8231 
8780 
9328 
9875, 

0422' 
0968 
ID|3 

2057 
2601 
3.44 
3687 
4229 


o533i  o589| 
.09.  1.47 
.649  .703 
2206!  2262 
2762I  2818 
33i8l  3373 
38731  3928 
4427  4482 
49S0!  5o36 
5533|  5588 
6oS5  6.40 
6636  6692 
7187  7242 
7737I  7792 
8286I  8341 
8835,  8890 


4770 

33.0 

585o! 
6389I 
6927 


4174 

47'6 

5256 

5796 

6335 

6874   , 

74.. I  7465 

7949'  8002 
908483'  8539 

9021  9074 

9556I  96.0 
910091I  0.44 

06241  0678 

1.58 

1690 


2222 
2753 
3284 


.21. 1 
.743 
2275 
2806 

333  r 


9383 

94^7 

99J0 

9983 

0476 

od3. 

1022 

.077 

.567 

1622 

21.2 

2166 

2655 

2710 

3199 

3253 

3741 

'4 

4878 

4283 

4824 

5364 

5418 

5904 

5958 

6443 

6497 

7o3D 

698. 

p.9 
8o56 

7573 
8..0 

8592 

8646 

9.28 

9.8. 

9663 

97.6 

0197 
073. 

02D. 

o]84 
.3.7 

1264 

1797 
2328 

i85o 

238. 

28591  29.3 
3390}  3443 


.042 
.6.3 
2.83 
2752 
332.1 
3888: 
4455' 

5022| 

5587 
6i52| 
67161 
7280' 
7842I 
64041 
89651 
9.')26 
••86' 
0645; 

.203, 

.760; 
23.7; 

2873' 
3429 
39S4 

4D38 
509.1 
5644  i 
6.95 

6747! 
7297| 

^96! 
8944! 
94921 
••39; 
o586| 
..3. 
.676' 
222.1 
2764 
3307! 
3849! 
43q«| 
49^2 
5472; 

60I2| 

655. 
7089; 
76261 
6.63, 
8609! 

92331 

9770 

o3o4! 
o838| 
.37.1 
.903 1 
2435 
2q66 
3496 


1099 
1670 
2240 
2809 
3377 
3945 
43.2 


8 


1.56 

1727 
22971 
28661 
3434I 
40021 
4369! 
5078;  5.33, 
5644:  5700 
6209!  6265; 
67731  6829I 
7i36|  7302I 
7808  7955; 
8460;  85.6 
902. i  9077 1 
9582 j  9b38: 
•14. 1  •.97I 
0700,  0736; 
.259  13.4I 
1816:  1872! 
2373|  24:9' 
2929  2985' 
3484|  3310, 
4039  4094 1 
45931  4648 
5i46|  5201 
5699'  5734 
623.1  63o6 
6802    6837 ; 


7352 
7902 
843. 
8999 
9547 
'94 


7407; 
83o6 

906  4: 

9602 
•.49' 
0640 1  0693 
..86'  1240 
173.1  .783, 

2275i  2320 
28.8,  2873, 

336.  3416' 


3904 
4445 
4986 

5326 

6066 
6604 
7143 
7680 
8217 
8753 


3958, 
4499' 

5o4o 
558o 
6. 19 
6658 
7196; 
7734 
8270 
880-:  1 


9289  9342  i 
9823:  9877J 
o338  04.. I 
0891  0944 
1424'  1477! 
19561  2009 
2488  2541  j 
30.9  3072 
3549  36o2 


I2l3{ 

1784I 
2354' 
2923 1 
349. 
4039' 

4623, 

5.92 
5p7 
6321 
6885 

7449 
80.. 
8573 
9.34 
9694 

•233 

0812 

1370 

I928I 

2484| 
3o4o 

3595 

4130 

4704I 

5237 

3S09' 

636. 1 
6912. 
.462' 
6012 
856 1 
9109 
9636 

•203 

0749 

.293 

1840: 

2384 
2927 

3470 
4012 
4553 
5094 
5634 
6173 
67.2: 
725o 
7787^ 
8324 
8860 
9396 
9930 
0464 

2o63: 
2594: 
3.25! 
3655 


1>. 


127. 
1841 
24.1 

2980 
3348 

4.13 

4682 

5248 
t8.3| 
6378 
6942 
73o5 
8067 
8629 
9190 
9730 
•309I 
0868! 
1426 
.983 1 
2340' 
3096 
365. 
42o5 
4759 
53.2 
5864 
64.6 
6967 
73.7 
6067 
86.5 


i328 
1898 
2468 
3o37 
3603 
4172 

tilt 

5870! 
6434! 

8.23 
8685 
9246 
9806 
•365 
0924 
1482 
2039 
2393 
3i5. 
3706 
426. 
4814 
5367 
5920 
6471 
7022 
7572 
8.22 
8670I 


9164!  9218 


•258: 

0804 

.349 

.8o.i 
24i8 
298.; 
3324 

4066; 

4607 1 
5.48; 

5688 
6227' 
6766' 
73o4j 
784. 
8378; 
8q.4' 
9449 

03.8 

io5. 

15841 

2..6j 

26471 
3.78, 


9766 

•3.2 

0859 

1404 
.948 
2492 
3o36 
3578 
4120' 
466. 

5202 

5742 
628. 
6820! 

73581 

m 

8967! 

93o3: 
••37! 
057. 
.Jo4 
.637 
2.69 
2700 
323. 
3761 


I 

u 

56 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 
55 
55 
55 
55 
55 
55 
55 
55 
55 
55 
55 
55 
55 
55 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
53 
53 
53 
53 
53 
53 
53 
53 


A    1ABL&    OF    LOGARITHMS    FROM     1    TO     lO,OOU. 


820 

0 

.  !  ,  1 

J     \ 

4  1  5    6  1 

1 
4184 

8  1 

9  I 

9i38i4 

3867;  39201 

39731 

4026;  4079  4 1 32 

4237]  4290 

821 

4343 

4396'  4449'  45o2| 

4555'  4608  4660!  4713 

4766;  48.9 

822 

4872 

4925  4977 

5o3o 

5o83j  5i36  5189'  5241 

52941  5347 

823 

5400 

5453  55o5 

5558 

56ii|  5664  5716,  5769 

5822  5875 

824 

5927 

5980  6o33 
65o7  6559 

6o85 

6i38:  6191  6243'  6296 

6349!  6401 

82c 

6454 

6612 

6664:  6717  6770';  6822 

6875,  6927 
74001  7453 

826 

6980 

7033  7083 

7i38 

7190,  72431  7295^  7348 

828 

7D06 

7558:  761 1 

7663 

7716;  7768I  7820:  7873 
8240'  8293  8345:  8397 

7925j  7978 

8c3o 

8o83  81 35 

8188 

845o!  8302 

829 

8555 

8607'  8659 

8712 

8764  8S16,  8869I  8921 

8973;  9026 

83o 

919078 

9i'Jo'  9183 

9235 

92871  9340 

9392!  9444 

9496,  9549 

83 1 

9601 

9653  9706 

9758 

9810  9S62 

9914'  99^^7 

••19  ••71 

832 

920123 

0176  0228 

0280 

o332i  o384 

0436  0489 

o54il  0593 

833 

0645 

0697  0749 

0801 

o853;  0906 

0938'  1010 

1062  1114 

834 

1166 

1218  1270 

l322 

i374  1426 

1478^  i53o 

i582,  i634 

835 

1686 

1738  1790 

1842 

1894'  1946 

1998J  2o5o 
25i8j  2570 

2102'  2 1 54 

836 

2206 

2258,  23l0 

2362 

2414  2466 

2^2  2674 

837 
838 

2725 

2777  2829 

2881 

2933  2q85 

3037:  3089 

3i4ol  3192 

3244 

3296,  3348 

3399 

345 1 i  35o3 

3555|  3607 

3658,  3710 

839 

3762 

3Si4  3865 

39.7 

3969'  4021 

40721  4124 

4176!  4228 

840 

924279 

4331!  4383 

4434 

4486:  4538 

4589 i  4641 

46931  4744 

841 

4796 

4848  48r,9 

49^1 

5oo3  5o54 

5io6  5i57 

5209'  5261 

842 

53i2 

5364  54 J  3 

5467 

55i8  5570 

562 1  i  5673 

3723;  5776 

843 

5828 

5879  5931 

5982 

6o34  6o85 

6i37  6188 

6240  6291 

844 

6342 

6394  6445 

6497 

6548  6600 

665ii  6702 

6754'  68o5 

845 

6857 

6908,  6950 
7422j  7473 

7011 

7062.  71 i4 

7i65!  7216 

7268I  7319 

846 

7370 

8o3^ 

7576  7627 

7678,  7730 
9191:  8242 

77811  7832 
8293  8345 

847 
848 

7883 

79351  7986 

80S8  8140 

8396 

8447 i  8498 

8549 

86oi|  8652 

8703.  8754 

88o5  8857 

849 

8908 

8959  9010 

9061 

9112!  9i63 

92i5!  9266 

9317!  9368 

85o 

929419 

9470  9321 

9572 

9623  9674 

9725,  9776 

9827:  9879 

85i 

9930 

oqSi.  ••32 

••83 

•i34'  •iSS 

•236|  •287 

•338!  ^389 

852 

930440  6491  0542 

0592 

0643  0694 

0745^  0796 

08471  0898 

853 

0949 

1000  io5i 

1102 

n53  1204 

1254'  i3o5 

i356j  1407 

854 

1458 

1 509'  i56o 

1610 

i66i  1712 

17631  1814 

1865  1915 

855 

1966 

2017:  2068 

2118 

2169  2220 

227IJ  2322 

2372  2423 

856 

2474 

2524:  2575 

2626 

26771  2727 

27781  2829 

2879!  2930 

858 

2981 

3o3ii  3o82 

3i33 

3i83  3234 

3285'  3333 

3386 

3437 

3487 

3538  3589 

3639 

3690  3740 

3791!  3841 

3892 

3943 

859 

^3993 

4044  4094 

4145 

4195!  4246 

4296;  4347 

4397 

4448 

860 

934498 

4549  4599 

46  5o 

4700I  4751 

4801!  4852 

4902 

4953 

861 

5oo3 

5o54  5 1 04 

5i54 

52o5  3255 

53o6:  5356 

5406 

5457 

862 

5507 

5558  56o8 

5658 

5709I  5759 

58091  586o 

5910 

5960 

863 

601 1  6061 i  61 11 

6162 

6212  6262 

63 13!  6363 

6413 

6463 

864 

6514!  6564:  66i4 

6665 

6715,  6765 

68j5 

6865 

6916 

6966 

865 

''Vit 

7066;  71 17 

7167 

7217I  7267 

7317 

7367 

7418 

7468 

866 

8019 

7568^  7618 
8069^  81 19 

7668 

7718,  7769 

7819 

7869 

7919 

7969 

868 

8169 

8219'  8269 

8320!  8370 

8420I  8470 

8520 

8570  8620 

8670 

8720  8770 

8820!  8870 

8920 

8970 

869 

9020  9070  9120 

9170 

9220  9270 

9320J  9369 

94iq 
9918 

746Q 

870 

939519  9369  9619 
94001 8j  0068  0118 

9719  9769 
0218;  0267 

9819  9869 

^8 

871 

0168 

o3i7 
0816 

0367 j  0417 

0467 

872 

o5i6j  o566  0616 

0666 

07161  0765 

o865j  0915 

0964 

873 

1014  1064  1114 

1 1 63 

i2i3  1263 

i3i3 

i362i  1412 

1462 

874 

i5u  i56i  1611 

1660 

1710!  1760 

1809 

1859'  1909 

,958 

875 

2008  2o58  2107 

2157 
2653 

2207  2256 

23o6^  2355;  24o5|  2455 

876 

25o4I  2554  26o3 

2702!  2752 

2801 i  285i:  2901 
3297  3346  3396 

2950 

t^l 

3ooo  3o49  3oo9 

3148 

3198;  3247 

3445 

3495  3544  3593 

3643 

3692,  3742 
4186:  4236 

3791'  3841  3890 
4285'  4335  4384 

3939 
4433 

879 

N. 

3989 
0 

4o38|  4o88|  4137 

,  1  2 

3 

4  i  5 

(3 

7 

1  8 

. 

^1 


A    TABLE    OF    LOGARITOMS    FROM    1    TO     10,000, 


16 


N.  1 

0   1  1 

2 

3  ;  4  1 

5  1  6  1  7 

8  ]  9  1 

T). 

49 

88o 

944483  4532 

458 1 

463 1,  4680'  4729I  4779'  4828  4877^  4927 

68i 

4976:  5o25j  5o74| 

5i24  5173  5222:  5272  5321  5370  5419 

49 

882 

5469I  55 18 

5567 

56i6  5665  57i5;  5764  58i3  5862^  5912 

49 

883 

59611  60101 

6059 

6108  6157  6207  6256  63o5|  6354'  64o3 

49 

884 

6452  65oi 

655i 

6600  6649'  6698  6747'  6796!  6845  6894 
7090  7'4o  7>89'  7238,  7287I  7336  7385 

49 

885 

6g43,  6992 
7434  74H3 

7041 

49 

886 

7532 

-i5Si^   763o  7679  7728,  7777i  7826  7875 

49 

88^ 

7924  7Q73 
841 3,  8462 

8022 

6070  81 19'  816S  8217  8266  83i5  8364 

49 

85ii 

856o  8609  8657;  8706I  87551  8804  88531  49 

889 

8902!  8951 
9 49390 1  9439 

8999 

9048  9097  9146  9195  9244'  9292  9341   49 
9536  9585  9634'  9683:  97311  9780;  9829!  49 

890 

94«8 

891 

9S78  9926 

9975 

••24  ••73  •i2ii  •170;  •219  •267,  •3i6'  49 
o5ir  o56o'  0608  o657|  0706  0754:  o8o3   49 

892 

9D0365  0414 

0462 

^^ 

o85i 

0900 

i386 

0949 

0907  1046  1095  1143!  1 192;  1240'  1289  49 
1483  i532;  i58o  1629  1677  1726  1776   4q 
196c'  2017  2066  2114!  2i63;  221 1 !  2260  48 
2453  25o2  255o  2599  2647I  2696  27441  48 

^i 

1 338 

I43DI 

895 

1823 

1872 

1920J 

896 

23o8. 

2356 

24o5 

897 

2792 1 

2841 

28i-;9! 

33731 

2938  2986  3o34  3o83 
34211  3470  35 1 81  3566 

3i3i|  3i8o  3228 

43 

898  1 

3276; 

3325 

36i5|  3663:  37 1 1 

48 

899 

376c 

38o8 

3856 

39051  3953 

4001 

4049 

4098  4146  4194 

48 

900 

954243 

4291 

4339! 

4387  4435 

4484 

4532 

458o  4628  4677 
5062'  5iio  5i58 

48 

901 

4725 

4773 

482 1 1  4S69'  49J8 

4966 

5oi4l 

48 

902 

5207 
5688 

5255 

53o3, 

535 1  i  5399 
5832  588o 

5447 

5495  5543  5S92'  5640 

48 

903 

5736 

5784' 

5928 

5976;  6024  6072I  6120 

48 

904 

61681 

6216 

6265' 

63 1 3  6361 

6400  6457 1 

65o5,  6553  6601 

43 

905 

6649 
712S 

6697 

6745 

6793  6840  6888;  6936' 

6984  7o32  7080 

48 

906 

7176 

7224 

72721  7320  73681  7416 

7464I  7512'  7559 

48 

907 

7607 

7655 
8i34 

7703 

775i!  7799 

7847  7894 
83251  8373 

7942 
8421 

7990  8o38 

48 

908 

8086 

8181 

8229'  8277 

8468  85 1 6 

48 

909 

8564 

8612 

8659 

8707 

8755 

88o3;  8850 

8898 

8946  8994 

48 

910 

9V4I 

9089 

9.37 

9185 

9232 

9280.  9328 

9375 

9423  9471 

48 

yll 

.  9518 

9566 

9614 

9661 

9709 

VM 

9804 

9852  9900 
•328  •376 

9947 

48 

912 

9995 

••42 

••90 

•i38 

•i8d 

•280 

•423 

48 

913 

960471 

o5i8 

o566 

061 3  0661 

0709 

0756 

0804 

o85i 

0899 

48 

94 

0946 

0994 

I04I 

1089!  n36 
i563  161 1 

1184'  I23l 

1270 
1753 

i326 

.374 

47 

9i5 

1421 

1460 
1943 

i5i6 

16581  1706 

1801 

1848 

47 

916 

1895 
2369 

1990 

2o38 

2o85 

2l32 

2180 

2227 

2275 

2322 

47 

918 

2417 

2464 

25ll 

2559 

2606 

2653 

2701 

2748 

2795 

47 

284i 

2890 

2937 

2985 

3o32 

3079 

3i26 

3.74 

3221 

3268 

47 

919 

33i6 

3363 

3410 

3457 

35o4 

3552 

3599 

3646 

3693 

3741 

47 

920  i  963788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4i65 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

449^ 

4542 

4590 
5o6i 

4637 

4684 

47 

922 

473 1 

4778 

4825 

4872 

i%l 

4966 

5oi3 

5io8 

5i55 

47 

923 

5202 

5249 

5296 

5343 

5437 

5484 

553 1 

5578 

5625 

47 

924 

5672 

5719 

5766 

58i3 

586o 

5oo7 
6376 

5954 

6001 

6048 

6095 

47 

923 

6142 

6189 
6658 

6236 

6283 

6329 

6423 

6470 

65i7 

6564 

47 

926 

661  I 

6705 

6752 

6799 

6845 

6892 

6939 

7408 

6986 

-033 

47 

928 

7080 

7127  7173 
7595  7642 
8062  8109 

7220 

7267 

73i4 

7361 

7454 

7501 

47 

7548 
8016 

768S 

778* 
8249 

7829 

7875 

7922 

7969 

47 

929 

8i56 

8296 

83431  8390 
8810!  88d6 

8436 

47 

930 

968483 

853o  8576 

8623 

8670;  8716 

8763 

8903 
.,369 

47 

931 

8950 

8996  9043 

9090 

9i36  9i83 

9229 

9276 

9323 

47 

931 

9416 

9463  9509 
!  99-8,  9973 

95561  9602  9649 

9742 

9789!  9B35 

47 

933 

9S82 

1  M,,!  MfeS  •ii4 

•207 

•254  •3or 

47 

934 

970347 

;  o3q3i  0440 

1  0486 

o533  o579 

0626 

0672 

0719  0765 
n83  122s 

46 

93D 

0812 

o85J 

5  0904 
1369 

1  0951 

0997  1044 
1461  i5o8 

\z 

1137 

46 

936 

1276 

i32: 

I4i5 

1601 

1647  16931  46 
:iIo'  2157   46 

$1 

1740 

i78< 

)  i832 

1879 

2388;  2434 

2018 

2064 

2203 

224t 

)  2293 

2342 

2481 

25271  2573  2619!  46  ! 
2989I  3o35  3082;  46  j 

939 

2666 

1  27121  2738 

2804I  285 1 !  2897 

2943 

IT. 

1    0 

L'_[.' 

3 

1  4 

1  ^ 

1  6 

r 

1  8 

1  9 

1  i>:i 

16 


K. 

940 
941 
942 
943 
944 

946 

9=i7 
948 
949 
900 
93, 
q5i 
953 
934 
955 
906 
957 
938 
959 
960 
961 
962 
963 
964 
965 
966 
967 
968 
969 
970 

97' 

972 
973 
974 

976 
977 
978 
979 
980 
981 
982 
983 
984 
983 
986 
987 
9S8 
989 
990 
991 
992 
993 
994 
995 
996 

997 
998 

999_ 


A  TAHLR 

OF 

LOQARrrUMS  FROM  1 

TO 

ib,UUU. 

0 

I 

2  1  3  1  4  1  5 

6 

7 

8  j  9 

D.  i 

46 

973128 

3174 

3220  3266'  33i3:  3359 

34o5 

34' I  3497  3543 

3390 

3636 

3682:  3728!  3774 

3820 

3866 

391 3i  3959  4oo5 

46 

4031 

4097 

4143  4189  4235'  4281 

4327 

4374!  4420,  4466 

46 

45i2 

4538 

4604'  465o  4696  4742 
5o64:  5no  5i56  5202 

4788 

4834 

4880 

4926 

46 

4972 

5oi8 

5248 

5294 

5340 

5386 

46 

5432 

5478 

5524'  5570I  56i6  5662 

5707 

5753 

5799 
6258 

5843 

46 

5891 

68?4 

5983'  6029  6075  6121 

6167 

6212 

63o4 

46 

6330 

6442'  6488  6533 

6579 

6625 

6671 

6717 
7175 

6763 

46 

6808 

6900  6946  6992 

7037 
74q5 

7083 

7129 

7220 

46 

7266 

73.2 

7358  74o3;  7449 

7541 

7586 

7632 
8089 

1678 
^135 

46 

977724 

7769 

7815  7861 

IZ  &  It 

8043 

46 

8181 

8226;  8272'  83i7 

85oo 

8546 

8591 

46 

8637 

86831  8728  8774 

8819  8865]  8911 

8956  9002 

9047 

46 

9093 

9.38 

9184  923o!  9273.  932ii  9366 

9412 

9437 

95o3 

46 

9348 

9394 

9639  9685j  9730  9776'  9821 

9867 

^6? 

9953 

46 

980003 

0049 
o5o3 

0094  0140  oiB5  023 1 '  0276 

o322 

0412 

43 

0458 

0349  0394'  0640  o685  0730 

077^ 

0821 

0867 

43 

0912 
1 366 

0957 

ioo3  1048 

1093  1 1 39'  1184 

1229 

1 683 

I3S5 

1 3  20 

43 

1411 

1456  i5oi 

i547  1592'  1637 

1728 

1773 

45 

1819 

1864 

1909  1954 

2000  2045 j  2090 

2 1 35 

2I8I 

2226 

45 

982271 

23i6 

2  362  2407 

2432  2497|  2543 

2588 

2633 

2678 

45 

27231  2769!  2814  2859 

2904  2949'  2994 
3356  340 1  i  3446 

3  040 

3o85 

3i3o 

45 

3175  3220J  3265  33io 

3491 

3536 

358i 

45 

3626  3671!  3716  3762 

3807  3852]  3B97 

3o42 

3987 

4o32 

45 

4077!  4122  4167  4212 

4257  43o2  4347 

4^92 

4437 

4482 

45 

4527  4372  4617  466: 

4707  4732,  4797 

4842 

4887 

4932 

45 

4977  5o22!  5067  5ii2  5i57  5202'  5247 

6292 

5337 

5382 

45 

5426  5471!  55i6  556i  56o6  565i!  56q6 

574. 

5786 

5830 

45 

5873;  5920,  5965  6010  6o55 

6100  6144 

6189 

6234 

6279 

45 

6324  6369'  64 1 3  645s 

65o3 

6548;  6593 

6637 

6682 

6727 

45 

98677J;  6817!  6861'  6906 

6951 

6996  7040 

7o85 

7i3o 

7175 

43 

7219  7264  7309:  7353 

7398 

7443]  7488 

7532 

7577 

7622 

40 

7666I  77 u  7756;  7800 
8n3|  8r57  8202'  8247 

7845 

7890  7934 
8336  838i 

7979 
8425 

8024 

8068 

43 

8291 
8737 

8470 

85i4 

45 

8559'  8604:  8648;  8693 
9003,  9049  9°94:  9i38 

87S2  8826 

8871 

8916 
9361 

8960 

45 

9i83 

9227!  9272 

93i6 

940  5 

43 

9450;  9494  9339  9583 
9805;  9939  9983|  ••28 

990339  o383  o428|  0472 
0783;  0827]  0871  0916 

991226  1270  i3i5|  i359 

9628 

9672 

9717 

9761 

9806 

9850 

44 

••72 

•1.7 

•161 

•206 

•25o 

•294 

44 

o5i6 

o56i 

o6o5 

o65o 

:fi^' 

0738 

44 

0960 

1004 

1049 

\t. 

1182 

44 

i4o3 

I448|  1492 
1890  1935 
2333  2377 

i58o 

1625 

44 

1669!  1713:  1758 

1802 

1846 

1979 

2023 

2067 

44 

211li  2l56  2200 

2244 

22S8 

2421 

2465 

25o9 

44 

2554  2598  2642 
2995:  3039;  3o83 

2686 

2730 

2774  2819 

2863 

2907 

2931 

44 

3127 

3172  32i6i  3260 

33o4 

3348 

3392 
3833 

44 

3436  3480  3524 

3568 

36i3 

3657  3701 

3745 

3789 

44 

38771  3921!  3965 
4317I  4361!  44o5 

4009 

4o53 

4097  4141 

4i85 

4229 

4273 

44 

4449 

4493 

4537]  458i 

4625 

4669  47 1 3 

44 

4737'  4801!  4845 

4889 

if. 

49771  5o2i 

5o65 

5 1 08 

5i52 

44 

5196'  5240'  5284!  5328 

54.6 

1  5460 

55o4 

5547 

5591 
6o3o 

44 

595635;  5679'  5723 

5767 

58ii 

5854 

1  5898 

5942 

5986 

44 

6074'  6117  6161 

62o5 

6249 

6293 

6337 

638o]  6424 

6468 

44 

65i2:  6555  6599 
6949  6993  7037 
7386  743o!  7474 

6643 

6687 

6731 

6774 

6818  6862 

% 

44 

7080 

7124 

7168 

7212 

7255  7299 
7692  7736 
8129  8172 

44 

7517 

7561 

7605;  7648 
8041  8o85 

7779 
8216 

44 

7823  7867  7910 
8239!  83o3  8347 

7934 

1% 

44 

8390 

8477!  8521 

8564!  8608 

8652 

44 

8693  8739'  8782!  8826 

8869'  8913]  8956 
93o3,  9348;  9392 

9000  9043 

9087 

44 

9i3r  9174  9218;  9261 

9435  9479 

'  9522 

■44 

9565  9609  9652  9696!  9739'  9783j  9826 

9870  9913 

1  9957 

1  43 

0 

i  • 

1   2 

.' 

1  4  1  5 

i  6 

7 

1  _«. 

1  9 

!  D.__ 

A  TABLE 

OF 

.  LOGARITHMIC 
SINES   AND   TANGENIS 


FOK   EVEKT 


DEGREE  AND  MINUTE 
OF  THE  QUADRANT. 


Kemark.  The  minutes  in  the  left-hand  column  of 
each  page,  increasing  downwards,  belong  to  the  de- 
grees at  the  top ;  and  those  increasing  upwards,  in  the 
right-hand  column,  belong  to  the  degrees  below 


18 

(0 

DPIGREES.)   A  TABLF 

OF  LOGARITHMIC 

M. 

0 

Sine 
0- 000000 

1).   1 

Cosine  !  I),  i  Tang. 

1). 

Cotang. 

60 

10.000000 

0 • oooooo 

Infinite. 

I 

6.468726 

5017-17, 

000000;  "00 

6-468776 

5017.17 

13-53627 i 

a 

764756 

2934-85 

000000  -00 

764756 

2984 

83 

285244 

3 

940847 

2082-31 

000000,  -00 

940847 

2082 

81 

0591 58 

57 

4 

7.065786 

1615-17I 
i3i9-68 
ni5-75 

0000001  -00 

7-065786 

i6i5 

•7 

12^984214 

56 

5 

162696 

000000 :  -00;   i6?696 

1819 
iii5 

H 

887804 
758122 

55 

6 

241877 

9.999999'  -01 

241878 

54 

I 

308824 

066-53 
852-54' 

999999  -oi 

308825 

^r 

591175 
638 1 83 

53 

366816 

999999  .01 

866817 

54 

52 

9 

417968 

762-63; 

999999 
999998 

-01 

417970 

762 

63 

582o3o 

5i 

10 

463725 

689-88 

-01 

468727 

689 

88 

586278 

12.494880 

457091 

5o 

II 

7.5o5ii8 

629-81! 

9.999998 

•01 

7.5o5i2o 

629 

81 

4q 
48 

12 

542906 

579-36 

999997 

•CI 

542909 

577672 

579 

33 

i3 

577668 

536-41 

999997 

•01 

586 

42 

422828 

47 

14 

609853 

499-38 

999996 

•01 

609857 

499 

',t 

890143 

46 

i5 

639816 

467-14 
438-81 

999996 

.01 

689820 

43J 

860180 

45 

i6 

667845 

999995 

.01 

667849 

82 

382i5i 

44 

\l 

694173 

4l3-72 

999995 

.01 

694179 

4i3 

73 

3o582i 

43 

718997 

391.35 

999994 

•01 

719004 

891 

36 

280997 
257516 

42 

^9 

742477 
764754 

371-27 

999993 

-01 

742484 

371 

28 

41 

20 

353-15 

999993 

-01 

764761 

35i 

86 

235289 

40 

21 

'■& 

336-72 

9-999992 

-01 

7-785951 

836 

73 

12.214049 
198845 

89 
38 

22 

321-75 

999991 

-01 

806155 

321 

76 

23 

825451 

3o8-o5 

999990 

-01 

825460 

3o8 

06 

174540 

37 

24 

843934 

295.47 

999989 

•02 

848944 
861674 

295 

49 

i56o56 

36 

25 

861662 

283.88 

999988 
999988 

•02 

288 

90 

188826 

85 

26 

878695 
895085 

273-17 

-02 

878708 

278 

18 

121292 

34 

11 

263-23 

999987 

-02 

895099 

263 

25 

104901 

38 

910879 

Ifl 

999986 

.02 

& 

254 

01 

089106 

32 

29 

926119 

999985 

.02 

245 

40 

078866 

3i 

3o 

940842 

237-33 

999983 

.02 

940858 

287 

35 

059142 

3o 

3i 

7.955082 

229-80 

9-999982 

.02 

7.955100 

229 

81 

1 2  -  044900 

ll 

32 

968870 

222-73 

999981 

.02 

968889 

222 

75 

08 1 1 1 1 

33 

982233 

216-08 

999980 

.02 

982253 

216 

10 

017747 

27 

34 

995iq8 
8-007787 

209-81 

999979 

.02 

995219 

lit 

88 

004781 

26 

35 

2o3.qo 

999977 

•02 

8-00780^ 

?3 

11-992191 

25 

36 

020021 

198-31 

999976 

.02 

020043 

198 

24 

ll 

081919 

193.02 

999975 

.02 

081945 

188 

o5 

23 

043 DO I 

188.01 

999973 

.02 

048527 

08 

956473 

22 

39 

054781 

183-25 

999972 

.02 

054809 

i83 

27 

945I9I 

21 

40 

065776 

178-72 

999971 

.02 

o658o6 

178 

74 

984194 

20 

41 

8-076500 

174-41 

9-999969 
999968 

.02 

8-076581 

174 

44 

11-928469 

9 1 8008 

:? 

42 

086965 

170-31 

-02 

086997 

170 

34 

43 

097183 

166-39 

999966 

•02 

097217 

166 

42 

002788 

n 

44 

107167 

162-65 

999964 

-03 

107202 

162 

68 

802797 
888037 

16 

45 

I  16926 

159.08 

Q99963 

•  03 

1 16968 

',U 

10 

i5 

46 

I  2647 1 

153.66 

999961 

.03 

126D10 

68 

873490 

4 

47 

i358io 

152.38 

999959 

•03 

i3585i 

l52 

41 

864149 

i3 

48 

144953 

149-24 

999958 

■o3 

I44Q96 

149 

27 

855004 

12 

49 

153907 

146-22 

999956 

.03 

1589^2 

146 

27 

846048 

II 

5o 

162681 

143.33 

999954 

.03 

162727 

143 

86 

887273 

10 

5i 

8-171280 
I797'3 
187985 
196102 

140-54 

9.999952 

.o3 

8.171828 

140 

57 

11.828672 

I 

52 

53 
54 

137-86 
1  135-29 
1  i32-8o 

999950 
999948 
Q99946 

.o3 
•  03 
.o3 

188086 
196156 

182 

84 

820287 
811964 
808844 

55 

204070 

i3o.4i 

999944 

•  03 

204126 

180 

-44 

ffi^ 

5 

56 

21 1895 
219581 

j  128.10 

999942 

-04 

211953 

128 

-14 

4 

u 

1  125.87 

999940 

-04 

219641 

125 

•  90 

780359 

772805 

3 

227134 

123.72 

999988,  -04 

227195 

123 

-76 

2 

59 

234557 

;  1 2 1 . 64 

999986,  -04 

284621 

121 

-68 

765379 

I 

60 

241855 

i   119-63 

9999841  -04 

241921 

119-67 

758079 

0 

Cosine 

1). 

Sine  j89°|  Cotang. 

D. 

_;rang. 

M. 

SINES  AND  TANGENTS 

(1  DEGllEE.) 

19 

}L. 

Sine  1 

D.      1 

CoBine  1 

D.  1  Tang. 

D.   1 

Cotang.  1 

0 

8.241855 

119.63 

9.999934 

•04  8-241921 

119-67 

11.758079!  60 

I 

249033 

117.68 

999932 

.041   249102 

117-72 

7 50898 1  59 
743833  58 

a 

256094 

ii5-8o 

999929 

-04   256i65 

113-84 

3 

363o42 

113.98 

999927 

-041   263ii5 

114-02 

736885 

U 

4 

269881 

112.21 

999923 

•04'   269936 

112-25 

730044 

5 

276614 
263243 

iio-So 

999922 

•04!   276691 

110-54 

723309 

55 

6 

108-83 

999920 

•04I   283323 

io8-87 

716677 

54 

I 

289773 

1 07 . 2 1 

999918 

•04   289836 

107-26 

710144 

53 

290207 

103-65 

999913 

.04   296292 

103-70 

703708 

52 

9 

302346 

104- i3 

999913 

•04:   3026J4 

104-18 

697366 

5i 

iO 

308794 

102-66 

999910 

•041   308884 

102-70 

691 116 

5o 

i: 

8.314904 

101-22 

9.999907 
999905 

•04  8.3i5o46 

101-26 

11-684954 

4Q 

13 

321027 

90.82 
98-47 

•04i    321122 

99-87 

678878 

48 

a 

327016 

332924 

999902 

•04   327114 
.o5   333025 

98-51 

672886 

47 

1  / 

97-14 

999899 

97-19 
95-90 

666973 

46 

15 

338753 

93-86 

999897 

•05   338856 

661144 

45 

i6 

344D04 

94-60 

999894 

•o5   344610 

94-65 

655390 

44 

\l 

35oi8i 

93-38 

999891 

.o5l   330289 

•o5;  333895 

93-43 

6497 ' ' 

43 

355783 

999888 

92^24 

644 io5 

42 

>9 

36i3i5 

999885 

•oSj   36i43o 

91-08 
82-85 

638510 

41 

20 

366777 

999882 

.05!   366895 

633 1 o5 

40 

31 

8.372171 

9-999879 
999876 

-o5  8-372292 

11.627708 
622378 

^ 

33 

377499 

87-72 

-o5   377622 

87-77 
86-72 

33 

382762 

86-67 

999873 

.05 

382889 

617111 

37 

34 

387962 
393 101 

85-64 

999870 

■  o5 

388092 
393234 

83-70 

611908 

36 

35 

84-64 

999867 

.05 

84.70 

606766 

35 

36 

398179 

83-66 

999''^64 

.o5 

398315 

83.71 

601685 

34 

11 

4o3i99 

82-71 

999861 

.o5 

403338 

82.76 
81-82 

596662 

33 

408161 

tu 

999838 

•  05 

4o83o4 

5867^7 

32 

39 

4i3o68 

999854 

.05 

4i32i3 

80-91 

3i 

3o 

417919 

79.96 

999831 

.06 

418068 

80 -02 

581932 

3o 

3i 

8-422717 

]U 

9.999848 

.06 

8.422869 
427616 

70-14 
78.30 

11-377131 

11 

33 

427462 

999844 

.06 

572382 
567685 

33 

432156 

77-40 
76-37 

999841 

.06 

4323i5 

77.45 

27 

34 

436800 

999838 

.06 

436962 
44i56o 

563c38 

26 

35 

441394 

75-77 

999834 

•06 

75.83 

558440 

23 

36 

445941 

74-99 

999831 

-06 

446110 

75-05 

553890  24 
549387  23 

U 

450440 

74-22 

999827 

-06 

45o6i3 

74-28 

454893 

73-46 

999823 

.06 

455070 

73-52 

544o3o  22 
540319!  21 

39 

459301 

72-73 

999820 

.06 

439481 
463849 

72-79 

40 

463665 

72-00 

9998161 

.06 

72-06 

536i5i|  20 

41 

8.467985 

71-29 

9.999812 

.06 

8-468172 

71.35 

11-531828 

:? 

43 

472263 

70-60 

999801 

.06 

472434 

70-66 

527546 
523307 
519108 

43 
44 

'^ 

69.91 
60-24 
68-59 

.06 
.06 

485o5o 

69.^8 
6?-65 

17 
16 

45 

484848 

999797 
999793 

•07 

514950 
5io83o 

i5 

46 

488963 

67-04 
67.31 
66.69 
66.08 

.07 

489170 
49J25o 

68-01 

14 

% 

493040 

999790 
999786 

-07 

67.38 

506750 

i3 

TI 

•07 

497293 

66.76 

502707 

\2 

49 

999782 

•07 

501298 
505267 

66-15 

498702 

11 

5o 

5o5o45 

65.48 

999778 

•07 

65-55 

494733 

11-490800 

486902 

10 

5i 

53 

516726 
52o55i 

64-89 
64-3i 

9-999774 
999769 

•07 
•07 

8.509200 
5 1 3098 

64-Q6 
64-39 

§ 

53 

63-75 

999763 

•07 

63-82 

483o39 

7 

54 

63-19 

999761 

•07 

5243^6 

63-26 

479210 
47^414 

6 

55 

524343 

62-64 

9997^7 

•07 

62-72 

5 

56 

528102 

62  -  !  1 

999753 

•07 

528349 

62.18 

4]i65i 
401920 

4 

U 

531818 

61.58 

999748 

•07 

532080 

61 -65 

3 

535523 

6i.o6 

999744 

•07 

535779 

6i-i3 

464221!   2 

59 

539186 

60.55 

999740 

•07 

a^I 

60-62 

;6o533  1 

6o 

543819 

60. 04 

999735 

-07 

60 -13 

456916  0 

1  Cosine 

D. 

1  Sine 

88° 

Cotang. 

D. 

Tang  1 

<!0 

(2 

DEGREES.)   A  TABLE  OF  LO( 

,&Rl 

niii 

i(J 

M.  , 

Sine  1 

D. 

Coriiue  1  D.  1 

Tang. 

Cotang.  I 

0 

8.542819 

60  •  04 

9.999735,  -071  8-543o£4l 

60-12 

11-456916'  60 

I 

546422 

59. 

55 

99973 1 i  -071 

3466911 

59- 

62 

453309!  5o 
449732'  58 

2 

im% 

59. 

06 

999726  -071 

550268; 

59. 

14 

3 

58- 

58 

999722 1  -081 

55381 7i 

58 

66 

446183  57 

4 

557054 

58- 

II 

999717:  -oS; 

557336! 

560828I 

58 

'9 

442664  56 

5 

56o54o 

57- 

65 

999713  -o^j! 

57 

73 

439172:  55 

6 

563999 
567431 

57 

19 

999.08  .08! 

564291 

57 

27 

433709  54 
432273  53 

I 

56 

74 

999704  -08 

567727 

56 

82 

570836 

56 

3o 

999699'  -08 

571137 

56 

38 

428863  5a 

9 

574214 

55 

87 

999694,  -08 

574520 

55 

95 

425480  5 1 

10 

8.580892 

55 

44 

999689;  .08 

577877 
8.58i2o8 

55 

32 

422123,  5o 

II 

55 

02 

9.999683,  .08 

55 

IC 

ir.418792;  49 
4154861  48 

12 

584193 

54 

60 

0996801  '08 

584514 

54 

68 

i3 

587469 

54 

19 

9996751  -08 

587795 

54 

27 

4i22o5  47 

14 

590721 

53 

ll 

999670  -08 

591031 

53 

87 

40S949  46 

i5 

593948 

53 

999665  -08 

594283 

53 

tl 

405717  45 
402 5o8  44 

i6 

597152 

53 

00 

999660:  -08 

597492 

53 

\l 

6oo333 

52 

61 

999655|  -08 

6oo6-7T 

52 

70 

3993231  43 

603489 
606623 

52 

23 

999rj50j  -08 

6o3839 

52 

32 

396161!  42 

19 

5i 

86 

999645   09 

606978 

5i 

94 

393022  41 

20 

609734 

5i 

49 

999640 

09 

610094 
8.6i3i89 

5i 

38 

3899061  40 

21 

8.612823 

5i 

12 

9.999635 

.09 

5i 

21 

II. 38681 1 1  39 
383738;  38 

22 

615891 
6i8q37 

5o 

76 

999629 

.09 

616262 

5o 

85 

23 

5o 

41 

999624 

.09 

619313 

5o 

5o 

380687!  37 

24 

621962 

5o 

06 

999619 

•  09 

622343 

5o 

i5 

377637!  36 
37464S1  35 

25 

624965 

49 

72 

999614 

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625352 

49 

81 

26 

627948 

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38 

999608 

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49 

47 

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27 

63091 1 
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40 
48 

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999597 
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29 

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16 

3 59907 j  3o 

11.3570181  29 

354147  28 

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8-642563 

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9.999581 

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84 

32 

645428 

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43 

999573 

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53 

33 

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12 

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47 

22 

351296  27 
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34 

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82 

999564 

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46 

91 

35 

65391 1 

46 

52 

99955H 

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61 

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36 

636702 

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22 

999553 

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657 ' 49 

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3i 

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11 

659475 

45 

92 

999547 

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63 

999341 

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49 

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67 

999475 

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8.696543 

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42 

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52 

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9-999463 

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8-697081 

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28 

11-302919   9 
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52 

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54 

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32 

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56 

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40 

97 

999431 

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41 

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999424 

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40 

85 

287917 
285465 

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2 

59 

40 

29 

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•  II 

716972 

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40 

283028 

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60 

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D 

06 

999404 

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AND  TANOKXI^ 

.  {3    DEGREES., 

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63 

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5o 

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999027 

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8-827011 

22 

9-999019 

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829874 
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08 

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23 

53 

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3o 

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166387:  6 

55 

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tl 

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83 

1645Z9I  5 

56 

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70 
57 

162679'  4 

ll 

3o 

43 

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3o 

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839956 

3o 

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3o 

45 

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59 

841774 
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17 

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32 

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60 

3o 

00 

998941 

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30-19 

;  Cosino 

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D 

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DEOKEKf 

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OF  LOOAUITHMIC 

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Sine   1 

D.   ! 

Cosine  j 

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D.   1 

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0 

8.843585 

3o-o5 

9-9q894i| 

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8-844644! 

30-19 

II -155356:  60  i 

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845387 
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29-02 
29-80 

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846455; 

30-07 

1 53545;  So 
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a 

998923 

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848260 

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3 

848971 

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4 

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29-55 

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29-70 

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29-58 

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6 

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29-46 

144597 

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29-19 

998878 

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29-35 

:42S'39 
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53 

857801 

29-07 

998869 

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858932 

29-23 

52 

9 

85g546 

28-96 

998860 

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29-11 

139314 

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10 

861283 

28-84 

998851 

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862433 

29  •  cc 

28-88 

13-567 

5o 

II 

8-863014 

28-73 

9-998841 

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8-864173 

11  135827 

ii 

13 

864738 

28-61 

998832 

•  15 

865906 

28.77 

134094 

i3 

866455 

28-50 

998823 

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867632 

28-66 

132368 

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14 

868i65 

28-39 
28-28 

998813 

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869351 

28-54 

1 30649 

46 

i5 

869868 

998804 

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871064 

28.43 

128936 

45 

i6 

871565 

28-17 

998795 

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872770 

28.32 

127230 

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17 

873255 

28-06 

998785 

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874469 

28-21 

1 2 553 1 

43 

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27-95 

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28-11 

123838 

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19 

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27-^6 

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877849 

28-00 

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20 

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998757 

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879529 

27-89 

120471 

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ai 

8-879949 

27-63 

9-998747 

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8-881202 

27-79 

II  118798 

l^ 

32 

881607 
883258 

27-52 

998738 

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27-68 

ii7i3i 

23 

27-42 

998728 

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884530 

27-58 

115470 

ll 

24 

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886542 

27-31 

998718 

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886 1 85 

27-47 

Ii38i5 

25 

27-21 

998708 

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887833 

27-37 

1 12167 

35 

26 

888174 

27-11 

9986^9 

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27.27 

1 io524 

34 

27 

889801 

27-00 

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89.112 

27-17 

108888 

33 

28 

891421 

26-90 
26-80 

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892742 

27-07 

107258 

32 

29 

893035 

99S669 

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26.97 
26.87 

105634 

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3o 

894643 

26-70 

998659 

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895984 
8-897596 

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3o 

3i 

8-896246 

26-60 

9-998649 

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26.77 

11-102404 

ll 

32 

8Q7842 

26-51 

998639 

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899203 

36.67 

100797 

33 

899432 

26-41 

998629 

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900803 

26-58 

099197 

27 

34 

901017 
902596 

26-31 

998619 

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902398 
903987 
905570 

26-48 

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26 

35 

26-22 

998609 

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25 

36 

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ll 

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22 

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40 

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25-56 

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25-74 

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9l5022 

25-47 

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918034 

25-65 

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44 

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25-38 

998516 

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25-56 

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25-29 

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25-47 
25-38 

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46 

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25-20 

998495 

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921096 

078904 
077381 

14 

% 

921103 

25-12 

998485 

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932619 

25-30 

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922610 

25-o3 

998474 

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924 1 36 

25-21 

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la 

49 

924112 

24-94 

24-86 

998464 

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925649 

25-13 

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5o 

925609 

998453 

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1   927156 

25-03 

072844 

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5i 

8-927100 

24-77 

9-998442 

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:  8-928658 

24-95 

24-86 

11-071342 

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52 

928587 
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24-69 

998431 

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069845 

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24-60 

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24-78 

068353 

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54 

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24-52 

998410 

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24-70 

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56 

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34-53 

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4 

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24-27 

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998366 
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24-45 

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937398 
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24-19 

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24-37 

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24-11 

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24 -30 

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998344 

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SINKS 

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8  940296 
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23 

25 

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9.998220, 

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8-957674 

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25 

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21 
21 

57 
5o 

997947' 
997933 

20 

988842 
990149 

2. 
2. 

78 
7' 

011.58 
ooo85i 

36 

989374 

2. 

44 

997922 

99145. 

21 

65 

008549 

24 

\l 

990660 

2. 

38 

997010 

997897 
997883, 

992750 

21 

56 

007250 
003955 

23 

99'9i3 

2. 

3. 

994045 

21 

32 

22 

39 

993222 

2. 

25 

995337 

21 

46 

004663 

2. 

4o 

994497 

21 

»9 

997872 

996624 

21 

40 

003376 

20 

41 

8.995768 

2. 

12 

9-997860 

8-997908 

21 

34 

11-002092 

;i 

42 

997036 

21 

06 

997847 

999188 

21 

27 

000812 

43 

998299 

2. 

00 

997835 

9 • 000465 

21 

21 

20-999535 
99S262 

17 

44 

99<)56o 

20 

l^ 

9978221 

00.738 

2. 

i5 

16 

45 

9 • 0008 1 6 

20 

997809! 

oo3oo7 

21 

S 

996993 

i5 

46 

002069 
0033 18 

20 

82 

997797! 
997784 

004272 

21 

995728 

.4 

% 

20 

.76 

005534 

20 

97 

994466 

i3 

004563 

20 

.-0 

99777 «i 

006792 

20 

tl 

993208 

12 

i^ 

oo58o5 

20 

•  64 

9977581 

008047 
009298 

20 

99.953 

II 

5o 

007044 
9-008278 

20 

-58 

9977451 

20 

80 

990702 

10.989454 

988210 

10 

5i 

20 

-52 

9.997732 

9.010546 

20 

74 

I 

52 

0095.0 

20 

-46 

9977'9; 
997706 

01 1700 
oi3o3i 

20 

68 

53 

010737 

20 

-40 

20 

62 

986969 

7 

54 

01.962 

20 

•34 

9976031 
997680 

014268 

20 

56 

985732 

6 

55 

0.3.82 

20 

-.1^ 

oi55o2 

20 

5i 

984498 
983268 

5 

56 

014400 

20 

997667 

016732 

20 

45 

i 

ll 

0.56.3 

20 

■17 

997654! 

017959 
019183 

20 

40 

982041 

016824 

20 

-.2 

99764 1 ' 

20 

33 

980817 

2 

59 

oi8o3i 

20 

-06 

997628 

020403 

20 

28 

978380 

I 

6c 

019235 
Coeine 

20 

-00 

9976141 

021620 

20 

23 

0 

__  1). 

Sine  [Sr-^ 

_Corang. 

P   _ 

Tar4r._ 

m7 

Hi 

(^ 

DEGREES.)   A  TABLE 

OF  LOGAKITiiMlt 

it. 

Sine 

D. 

Cosine 

). 

Tang. 

J;,  n 

Cotang. 

60 

0 

9.019235 

20-00 

9-997614 

I2 

9-021620 

20-23 

10-978380 

I 

020435 

•9 

o5 

997601 

22 

02834 

20 

17 

9771661  5*1 
975q56  58 

a 

021632 

19 

B9 

997588 

22 

024044 

20 

3 

022825 

19 

84 

997374 

22 

025231 

20 

06 

974749'  57 

4 

024016 

19 

78 

9973611 

22 

026455 

20 

00 

973543!  56 

5 

025203 

19 

73 

997547 

22 

021655 

'9 

93 

972345  55 

6 

026386 

J9 

67 

997534 

23 

028852 

'9 

90 

971148;  ^4 

I 

027367 

19 

62 

997320 

23 

o3oo46 

'9 

85 

969954  53 

028744 

19 

57 

997307 

23 

o3i237 

'9 

79 

968763 

52 

9 

029918 

'9 

5i 

997493 
997480 

23 

032425 

'9 

74 

967575 

c' 

10 

031089 

19 

47 

23 

o336o9 

'9 

69 

966391 

5o 

II 

9-032257 

19 

4i 

9-997466 

23 

9-034791 

19 

64 

10-965209 

^ 

12 

033421 

19 

36 

997452 

23 

035969 

19 

58 

964031 

i3 

034582 

'9 

3o 

997439 

23 

037144 

19 

53 

962856 

47 

14 

035741 

>9 

25 

9974231 

23 

o3S3i6 

>9 

48 

961684 

46 

i5 

036896 

'9 

20 

9974  HI 

23 

039485 

'9 

43 

96051 5 

45 

i6 

o38o48 

'9 

i5 

9973971 

23 

040631 

'9 

38 

95q349 
938187 

44 

n 

039197 

•9 

10 

997383 

23 

041813 

19 

33 

43 

i8 

040342 

;? 

o5 

997369 

23 

042973 

'9 

28 

957027 

42 

19 

041485 

99 

99733-) 

23 

044 1 3o 

19 

23 

955870 

41 

20 

042625 

18 

94 

997341 

23 

045284 

'9 

18 

954716 

40 

21 

9-043762 

18 

89 

9-997327 

24 

9-046434 

19 

i3 

10-953566 

? 

22 

044895 

18 

^4 

9973 1 3 

24 

047582 

'9 

08 

952418 

23 

046026 

18 

79 

697299 
9972S3 

24 

048727 

'9 

o3 

951273 

37 

24 

047154 

18 

73 

24 

049869 

18 

98 

95oi3i 

36 

25 

048279 

18 

70 

997271 

24 

o5ioo8 

18 

t 

lt?,l 

35 

26 

049400 

18 

65 

997257 

24 

032144 

18 

34 

11 

o5o5i9 
o5i63! 

18 

60 

997242 

24 

053277 

18 

84 

946723 

33 

18 

55 

997228; 

24 

034407 

18 

79 

9/45593 

?' 

^9 

032749 
053859 

18 

5o 

997214; 

24 

035535 

18 

74 

944465 

3i 

3o 

18 

45 

997 '99 

24 

036639 

18 

^? 

943341 

3o 

3i 

■9  -  054966 

iS 

41 

9.9971831 

24 

9-057781 

18 

65 

10-942219 

It 

32 

056071 

18 

36 

997570 

24 

038900 

.8 

^ 

94 1 1 00 

33 

058271 

18 

3i 

997156 

24 

060016 

.8 

930984 
938870 

n 

34 

18 

27 

997'4i| 

24 

061 i3o 

.8 

5i 

26 

35 

059367 

18 

22 

997' 27 i 

24 

062240 

18 

46 

937760  25 

36 

060460 

iS 

17 

997112 

24 

063348 

18 

42 

936652 

24 

ll 

o6i55i 

18 

i3 

997008, 
997083, 

24 

064453 

18 

37 

935547 

23 

062639 

18 

08 

25 

o65556 

18 

33 

9344  a 

22 

59 

063724 
064806 

18 

04 

997068; 

25 

066655 

18 

28 

933345 

21 

40 

17 

99 

997053 

25 

067752 

18 

24 

932248 

20 

41 

9-065885 

17 

q4 

9-997.)39J 

25 

9-068846 

18 

'9 

io-93ii54 

>9 

42 

06696: 

17 

00 

9970241 

25 

069938 

18 

i5 

930062 

18 

43 

o68o36 

I  / 

^6 

9.;7009: 

25 

071027 

18 

10 

928973 

927S87 

'7 

44 

069 1 07 

17 

81 

906994 

25 

072113 

18 

06 

16 

45 

070176 

J7 

77 

9<i6<)79' 

25 

073197 

18 

02 

926803 

i5 

46 

071242 

17 

72 

<;')6c)64; 

25 

074278 

17 

97 

925722 

14 

4-' 

072306 

17 

68 

996049 

25 

075356 

17 

93 

924644 

i3 

48 

073366 

17 

63 

;96934: 

25 

076432 

17 

h 

923568 

12 

49 

074424 

17 

i? 

^96919 

25 

0775o5 

17 

84 

922495 

5o 

075480 

17 

996904, 

25 

078576 

17 

80 

921424 

10 

5i 

9-076533 

'  n 

5o 

9-C96889! 

25 

9-079644 

17 

76 

10-920356 

I 

5a 

077583 

17 

46 

996S74! 

25 

0S07 1 0 

17 

72 

919290 

53 

54 

078631 
079676 
080719 

17 
17 

42 

38 

996858; 
996843! 

23 
25 

081773 
oS:833 

17 
17 

67 

63 

918227 
9.17167 

I 

55 

17 

33 

996828 

25 

083891 

17 

59 

916109 
9i5o53 

5 

56 

081759 

17 

29 

996812' 

26 

084947 

17 

55 

4 

57 

082797 
083832 

17 

23 

996797 

26 

086000 

17 

5i 

914000 

3 

53 

■  17 

21 

996782 

26 

087050 

17 

47 

912950 

2 

6^^ 

084864 
085894 

17 
17 

\l 

996766 

996731 : 

26 
26 

3^ 

088098 
08914-i 

_Cotar^._, 

17 
17 

43 
38 

QI1902 
910856 

I 
0 

L__. 

Cofiit;e 

u 

. 

Sine  _  8 

D 

JL' 

BINES  AND  TANUKNTS.   (7  DEGREES. 

;            ^ 

o 

Sino 

D. 

Coahio 

D. 

1  Tanjf. 

IX 

Cotanjf.  1 

9.085894 

1713 

9.996751 

•  26 

1  9-089144 

17-38 

10.910856;  60 

a 

086922 

086970 

17.09 
17.04 

996733 
996720 

.26 
.26 

i   090181 
.   091228 

17-34 
17-30 

907734;  57 

3 

17-00 

996704 

.26 

092266 

17.27 

4 

089990 

16-96 

996688 

•26 

oo33o2!  17-22 

906698  5b 

5 

091008 

16-93 

996673 

'  .26 

!   09 i 336 

17-19 
17-15 

905664'  55 

6 

092024 

16.88 

996657 

.26 

1   095367 

904633  54 

I 

093037 

16.84 

996641 

•26 

096395 

17.11 

9o36o5,  53 

094047 

16.80 

996625 

-26 

097422 

17.07 

902578;  5a 

9 

093056 

16.76 

996610 

•  26 

098446I  17 -03 

90I534  5i 

ro 

C96062 

16.73 

996394 

.26 

099468 

1  '6-99 

0005321  5o 

io.8995i3|  49 

89.W  48 

II 

9.097065 

16.68 

9.996578 

•27 

9-100487 

1  16-93 

12 

098066 

16.65 

996562 

•27 

1 0 1 5o^ 

1  '^-g' 
16-87 

i3 

099065 

16.61 

996546 

•27 
.27 

I025f9 

897481 1  47 
896468  46 

14 

100062 

16.57 

996530 

103532 

16-84 

i5 

ioio56 

16.53 

9965 1 4 

•27 

104542 

16-80 

.  895438;  45 

i6 

102048 

16.49 

996498 
996482 

•27 

I03550 

16.76 

894450  4i 

]l 

io3o37 
104025 

16-43 

•37 

106556 

1  •6-72 

893444  43 

'.6-41 

996465 

•27 

107559!  16-69 

892441  42 

'9 

io5oio 

16-38 

996449 
996433 

•27 

io856o 

16-65 

891440  41 

20 

105992 

16-34 

•27 

109359 

16-61 

890441  40 

21 

9-106973 

i6-3o 

9-996417 

•27 

9-1 ro556 

16-58 

10.889444,  39 
8S§449  38 

22 

10795 1 
io'3927 

16-27 

996400 

•  27 

I r r55i 

16-54 

23 

16-23 

996384 

•27 

112343 

i6-5o 

8874371  37 
886467!  36 

24 

109901 

16-19 
16-16 

996368 

•27 

113533 

16-46 

25 

110873 

996351 

•27 

114331 

16. 43 

885479!  35 
884493,  34 

26 

111842 

16-12 

996335 

•  27 

1 1 5507 

16-39 

u 

112809 

16-08 

996318 

:ll 

1 16491 

16.36 

883509I  33 
882528,  32 

113774 

i6-o5 

99o3o2 

1 17472 

16.32 

29 

114737 

115698 

9. 1 16656 

i6-oi 

996283 

-28 

11S432 

16.29 

881548  3i 

3o 

15-97 

996269 

•28 

1 19429 

16-23 

880571  3o 

3i 

i5-94 

9-996232 

.28 

9- 120404 

16.22 

10.879596  29 
878623  28 

32 

117613 

15.90 

996235 

.28 

121377 

16.18 

33 

II 8567 

15.87 

996219 

.28 

122348 

i6-i5 

877632!  37 

34 

119519 

i5-83 

996202 

•28 

123317 

1611 

876683;  26 

35 

1 20469 

i5-8o 

996185 

•28 

124284 

16-07 

875716'  25 

36 

121417 

15-76 

9961 68 

.28 

125249 

16.04 

874751!  24 

ll 

122362 

15-73 

996131 

.28 

126211 

16.01 

873789!  23 

872828:  22 

i233o6 

15.69 

996134 

•28, 

i28i3o 

i5-97 

39 

124248 

15-66 

996117 

•28 

15.94 

871870:  21 

40 

125187 

9-I26I2D 

1 5.62 

996100 

.28 

1 29087 

i5.o. 
15.87 

870913  20 

41 

15.59 

9 -996083 

•29' 

9-i3oo4i 

10.869939'  19 

42 

127060 

«5-56 

996066 

•29 

1 30994 

15.84 

868o56  17  1 
867107  16 
866161 i  i5 

43 

127993 

i5.52 

996049 

•29 

133839 

15.81 

44 
45 

128925 
129854 

;i:s 

996032 
996015 

•29 
•29 

.5.77 
15-74 

46 

130781 

15.42 

99399S 

.291 

134784 

15.71 
15.67 

865216'  ri 

% 

131706 

15.39 

993980 

•29! 

135726 

864274  1 3 
863333  12 

1 32630 

i5.3d 

993963 

•29,' 

136667 
137605 
138542 

15.64 

49 

i3355i 

15-32 

993946 

.29I 

i5.6i 

862395  11 

5o 

134470 

9.135387 

i363oi 

15.29 

993928 

•29' 

15-58 

861458!  10 

5i 

5s 

15.23 
15-22 

9.993011 
993^4 

.29 
.20 

9-139476 
140409 

15.55 
i5.5i 

10-860524  9 
859591;  8 
838660'  7 
857731;  6 

53 

137216 
i3gi28 

15.19 

99">'<76|  -20 

141340' 

15.48 

54 

i5.i6 

99^^59, 

•29 

142269' 
143196 

15.45 

55 

139037 

l5.|2 

99>'^4i; 

-29 

15.42 

856804  5 

56 

139944 
i4o«5o 

13.09 

995S23 

•29 

1441 21' 

15.39 

855879   4 
854956  3 

u 

l5.o6 

995806 1 

•29 

143044 

15.35 

141754 
142655 

l5-o3 

995788  .29 

143966 
146885 

13-32 

834034  i 

59 

I5.00 

99^77'  -29 

15-29 
15.26 

833ii5   I 

6o^ 

143555 

14-96 

9957531  -29 

147803 

852197   0  1 

Ci^irie 

_J>:_i 

Siiie  1^30 ! 

C0t!Ulg._| 

_l^-._.i 

Tiiiig.  J_il.  J 

ZG 

{^ 

DEGREES.)   A  TABLE 

OF  LOGARITHMIC 

TIT" 

0 

Sine 

D. 

Cosino 

D. 

Tan-. 

B. 

Cctang. 

9-143555 

14-96 

9.995753 

".3o 

9-147803 

13.26 

IO-852I07 
851282 

6o" 

I 

144453 

14 

93 

995735 

.30 

148718 

i5 

23 

u 

5 

145349 

14 

90 

995717 

.30 

149632 

i5 

20 

85o368 

3 

146243 

14 

87 

995699 

-3o 

1 5o544 

i5 

17 

849456 

u 

4 

i47i36 

14 

84 

995681 

-3o 

1 5 1454 

i5 

14 

848546 

5 

148026 

14 

81 

995664 

.30 

152363 

i5 

II 

847637 

55 

6 

1489 1 5 

14 

7S 

995646 

-3o 

153269 

i5 

08 

846731 

54 

I 

140802 

14 

75 

995628 

•3c 

154174 

i5 

o5 

845826 

53 

1 50686 

14 

72 

995610 

-3o 

1 55077 

i5 

02 

844923 

5a 

9 

i5i569 

14 

69 

995591 

•3o 

155978 

14 

99 

844022 

5i 

10 

1 5245 1 

14 

66 

995573 

-30 

156877 

14 

96 

843123 

5o 

II 

5-i5333o 

14 

63 

9-995355 

-3o 

9-137773 

14 

93 

10-842225 

^^ 

12 

i542o8 

14 

60 

995537 

-3o 

158671 

14 

?7 

841329 
840435 

i3 

i55o83 

14 

57 

995019 

-30 

1 59565 

14 

ii 

14 

i55n57 
1 56830 

14 

54 

995501 

-3i 

160457 

14 

84 

839543 

i5 

14 

5i 

995482 

•  31 

161347 

14 

81 

838653 

0 

i6 

157700 

14 

48 

995464 

-31 

162236 

14 

79 

837764 

44 

\l 

158569 

14 

45 

995446 

.3i 

1 63 1 23 

14 

76 

836877 

43 

159435 

14 

42 

995427 

-31 

164008 

14 

73 

835992 

42 

'9 

i6o3oi 

14 

39 

995409 

-3i 

164892 

14 

70 

835io8 

41 

20 

161 164 

14 

36 

995390 

•  31 

165774 

14 

67 

834226 

40 

21 

9-162025 

14 

33 

9-995372 

•  31 

9-166654 

14 

64 

10-833346 

U 

3i 

162885 

14 

3o 

995355 

•  3i 

167532 

14 

61 

832468 

-.3 

163743 

14 

27 

995334 

-3i 

168409 

14 

58 

83 159 1 

37 

24 

164600 

U 

24 

995316 

-3i 

169284 

14 

55 

830716 

36 

25 

165454 

14 

22 

995297 

-3i 

170157 

14 

53 

829843 

35 

26 

1 66307 

U 

'9 

995278 

-3i 

171029 

14 

5o 

828971  34  1 

27 

1 67  09 

14 

16 

995260 

•  31 

171899 

14 

47 

828101 

33 

28 

168008 

14 

i3 

995241 

-32 

172767 

14 

44 

827233 

32 

-jg 

168856 

14 

10 

995222 

-32 

173634 

14 

42 

826366 

3i 

3o 

169702 

14 

07 

995203 

-32 

174499 

14 

39 

825501 1  3o 

3i 

9-170547 

14 

o5 

9-995184 

.32 

9-175362 

14 

36 

io.824638i  20 
823776  28 

32 

171389 

14 

02 

9951 65 

-32 

176224 

14 

33 

33 

172230 

i3 

99 

995146 

•32 

177084 

14 

3i 

8229161  27 

34 

173070 

i3 

96 

995127 

•32 

177942 
178799 

14 

28 

8220581  36 

35 

173908 

i3 

94 

995108 

.32 

14 

25 

821201!  25 

36 

174744 

i3 

11 

995089 

.32 

179653 

14 

23 

820345  24 

u 

175578 

i3 

995010 
995o5i 

.32 

i8o5o8 

14 

20 

819492  23 

17641 1 

i3 

86 

.32 

i8i36o 

14 

n 

818640  22 

39 

177242 

i3 

83 

995o32 

-32 

182211 

14 

i5 

817789  21 

40 

178072 

i3 

80 

9950 1 3 

•32 

iS3o59 

14 

12 

816941  20 

41 

9-178900 

i3 

77 

9.99499^ 

-32 

9.183907 

14 

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990538 1   . 

44 

324358 

10-41 

675643 

5 

56 

9-96 

99o5iil   . 

45 

3249S3 

10-40 

675017 

4 

i 

316092 

9-94 

990485    . 

45 

325607 

10-39 

tl^ 

3 

316689 

9  93 

990458,   . 

45 

32623. 

.0.37 

■2 

S 

317284 

9-91 

99043 1     • 

45 

326853 

10-36 

673 1 /,7 
672535 

I 

6o 

317879 

9.90 

990404    • 

45 

327475 

10-35 

0 

Cosine 

D. 

Sine      7 

8° 

CotaiiS'. 

-    D. 

Tang. 

30 

(12  DEGREES.)   A 

TABLE  OP  LOGARITHMIC 

M. 

0 

j  Sine 

1). 

j  Cosine  |  T>. 

i  Tang. 

D. 

Cotanar. 

60 

9.377879" 

9.90 

9-99o4oi  -45 

'  9-327474 

10.35 

10-672526 

I 

318473 

990378  -451   3:8095 

10.33 

671905 

U 

2 

319066 

9.87 

99o35ii  .45 

328715 

10-32 

671285 

3 

319658 

?-86 

990324I  -45 

329334 

10-30 

670666 

u 

4 

320249 

9.84 

9902071  .45 

329953 

10-29 

670047 

5 

320840 

9-83 

990270'  '451   33o57o 

10.28 

669430 
6688 1 3 

55 

6 

321430 

9-82 

9902 i3  .45 

;  331187 

10-56 

54 

I 

322019 

9-8o 

990215;  -45 

!   33i8o3 

IO-25 

668197 
667 58 2 

53 

322607 

9-79 

900 1 88,  .451   332418 

10-24 

52 

9 

323194 

!  523780 

9-77 

990 1 6 1 

1  .451   333o33 

10-23 

666967 
666354 

5i 

ic 

9.76 

990134 

i  -45 

333646 

10-21 

DO 

ii 

9-324366 

9.75 

9-990107 

•46 

;  9-334259 

10-20 

10-665741 

it 

13 

324950 

9.73 

990079 

i  -46 

334871 

10- 19 

665129 
6645.8 

i3 

325334 

9-72 

990002 

1  -46 

335482 

10-17 
10-16 

il 

14 

326117 

9-70 

990025 
989997 

.46 

336093 

663907 

i5 

326700 

li^ 

.46 

J36702 

.o-i5 

663298 

45 

i6 

327281 

989970 

.46 

337311 

io-i3 

662689 

44 

11 

327862 

9-66 

989942 

•46 

337919 

10-12 

662081 

43 

328442 

9-65 

989015 
989887 

.46 

338527 

10-11 

661473 

42 

19 

329021 

9.64 

.46 

339.33 

10-10 

660867 

41 

20 

329699 

9-62 

989860 

.46 

339739 

io-o8 

660261 

40 

21 

9-330176 

9-6i 

9-989832 

.46 

9-340344 

10-07 

10-659656 

It 

22 

33oi53 

9-60 

989804 

.46 

340948 
34i552 

10 -06 

659052 
658448 

23 

33i3:9 

Q-58 

989777 

.46 

10-04 

11 

24 

331903 

9.57 

989749 

•47 

342155 

io-o3 

657845 

25 

332478 

9-56 

989721 

•47 

342757 

10-02 

657243 

35 

26 

333o5i 

9-54 

989693 

•47 

343358 

10-00 

656642 

34 

11 

333624 

9-53 

989665 

•47 

343958 
344558 

9-99 
9-98 

656o42 

33 

334195 

9.52 

989637 

•47 

655442 

32  - 

29 

334^66 

9 -So 

989609 

•47 

345i57 

9-97 

654843 

3 1 

3o 

33533- 

9-49 

989582 

•47 

345755 

9-96 

654245 

3o 

3i 

9-335906 

9-48 

9.989553 

•47 

9-346353 

9-94 

.0-653647 

It 

32 

336475 

9-46 

989525 

•47 

346949 
347545 

9-93 

653o5. 

33 

337043 

9-45 

989497 

•47 

9-92 

652455 

U 

34 

337610 

9.44 

989469 

•47 

348141 

9-91 

65i859 
65.265 

35 

338176 

9-43 

989441 

•47 

348735 

9-90 

25 

36 

338742 

9-41 

9894.3 

•47 

349329 

9-88 

650671 

24 

il 

339306 

9-40 

9S9384 

•47 

349922 
35o5i4 

lii 

650078 

23 

38 

339871 

9-39 

989356 

•47 

649486 
648894; 

22 

39 

340434 

9.37 

989328  .47 

35iio6 

9-85 

21 

40 

340996 
9-341558 

9-36 

^8^300  .47 

351697 

9-352287 

352876 

353465 

9-83 

6483o3 

20 

41 

9-35 

0  989271 

•47 

9-82 

io-6477'3| 

',t 

42 

342119 

9-34 

'  989243 

•47 

9-81 

6471241 

43 

342679 

9-32 

989214 

•47 

9-80 

646535;  17 

645047 i  16 

44 

343239 

9-3i 

989186 

•47 

354053 

Q-79 

45 

343797 

9 -30 

9S9157 

•47 

354640 

9-77 

645360'  .5 

46 

344355 

9-29 

989128 

.48 

355227 
3558i3 
356398 
356982 
357566 

9-76 

644773;  14 

% 

344912 
345469 

9.27 
9-26 

989100 
989071 

:|- 

9-75 
9-74 

644187:  1 3 

643602   .2 

i9 

346024 

9-25 

989042 

•48 

9-73 

643-18  :i 

5o 

346579 

9-24 

989014 

•48 

9-71 

642434  10 

5i 

9  347 '34 

9-22 

9-988985  .48: 

9.358149 

9-70 

io.64i85i,  9 
641269  § 

5i 

347^87 

9-2' 

988956  .48: 

35873. 

9-6o  1 

9.68  ; 

53 

348240 

9-20 

988927'  -48 

359313; 

640687  7 
640107  6 

54  ' 

348792 

9-19 

988898.  .48: 

359893 

9.67 

55  1 

349343 

9-17 

988869;  .48: 

360474 

9-66 

639326   5 

56  ; 

349893 

9.16 

988840  .48; 

36io53 

9-65 

63^947   4 

i1 

350443 

9-15 

9888.11  .49' 

36.632 

Q-63 

638368   3 

5b   ' 

350992 

9-14 

988782  .49' 

3622 10 

9-62 

637790   J 

59 

35i34o 

Q.l3 

988753  .49' 

362787, 

9-61 

647213   1 

60 

352088  1 

9-II 

988724  .49' 

363364 

9-60 

636636  u  1 

Cosine  ' 

D. 

Sine   770 

Cotnng. 

T). 

__TaLiS^l 

M.| 

8INES  AND  TANGENTS. 

(13  DKORKKS. 

) 

3 

'.^ 

Sine 

1). 

Cosine 

D. 

Tan?. 

1). 

Cotanj;, 

0 

9.352088 

9-11 

"^988724 

—49 

9-363364 

9-60 

10  036636 

"6^ 

I 

352635 

9-10 

988695 

.49 

363940 

9 

S 

636o6o 

5? 

2 

353i8i 

9. 09 
908 

988666 

.49 

364^15 

9 

635485 

3 

353726 

988636 

.49 

365otio 

9 

U 

6049.0 

57 

4 

354271 

9.07 

9-03 

988607 

.49 

365664 

9 

634J36 

56 

5 

354815 

9H8578 

.49 

366237 

9 

54 

633763 

55 

6 

355358 

904 

988548 

.49 

366810 

9 

53 

633.901  54 

2 

355901 

903 

988519 

.49 

367382 

9 

52 

6326i8'  53 

356443 

9-02 

988489 

.49 

367953 
368524 

9 

5i 

632047  52 

9 

356oH4 

357524 

9.358064 

901 

h 

988460 

.49 

9 

5o 

631476 

5i 

10 

II 

988430 
9.988401 

.49 
•49 

9.369663 

9 
Q 

tt 

630906 
io.63o337 

5o 

it 

12 

3586o3 

8-97 

988371 

•49 

370232 

9 

46 

629768 

i3 

35914 1 

8-96 

988342 

•49 

370799 

9 

45 

629201 
628633 

47 

14 

359678 

8.95 

988312 

.50 

371367 

9 

44 

46 

i5 

36021 5 

8.93 

988282 

.50 

371933 

9 

43 

628067 

45 

i6 

360752 

8-92 

988252 

.50 

372499 

9 

42 

627501 

44 

\l 

361287 

8-91 

988223 

.50 

373064 

9 

41 

626936 

43 

361822 

8-90 

988193 

.50 

^?^^P 

9 

40 

626.171 

42 

»9 

362356 

8-80 

988163 

.50 

9 

^ 

625807 

41 

20 

362889 

8-88 

988133 

.50 

3747^6 

9 

623244 

40 

21 

9-363422 

8-87 

9.988103 

.50 

9.375319 

9 

ll 

10-624681 

ll 

22 

363954 

8-85 

988073 

.50 

375881 

9 

624. .9 
623558 

23 

364485 

8.84 

988043 

•50 

376442 

9 

34 

37 

24 

3650 16 

8-83 

988013 

.50 

377003 

9 

33 

622997 
622437 

36 

25 

365546 

8.82 

987983 

.50 

377563 
378122 

9 

32 

35 

26 

366075 

8-81 

987953 

.50 

9 

3i 

621878 

34 

11 

366604 

8.80 

tt^ 

.50 

378681 

9 

3o 

62.3.9 

33 

367i3i 

8-79 

.50 

379239 

9 

ll 

620761 

32 

29 

367659 
368 1 85 

8-77 

987862 

.50 

'^Itl 

9 

620203 

3i 

3o 

8-76 

987832 

.51 

9 

ll 

619646 

3o 

3i 

9-36871 1 

8.75 

9.987801 

.51 

9-380910 

9 

10-619090 
618534 

ll 

32 

369236 

8-74 

987771 

.51 

381466 

9 

25 

33 

369761 

8.73 

987740 

.51 

382020 

9 

24 

617980 

27 

34 

370285 

8.72 

.  987710 

•  51 

382575 

9 

23 

617425 

26 

35 

370808 

8.71 

987679 

•51 

383129 

9 

22 

616871 

25 

36 

37i33o 

8-70 

987640 
987618 

.51 

383682 

9 

21 

6i63i8 

24 

ll 

371852 

.51 

384234 

9 

20 

615766 

23 

372373 

8.67 
8-66 

987588 

.51 

384786 

9 

\l 

6l32l4 

22 

39 

372894 

987557 

•51 

385337 
385888 

9 

614663 

21 

4o 

3734.4 

8-65 

987526 

•51 

9 

\l 

614112 

20 

41 

9.373933 

8-64 

9.987496 

.51 

9.386438 

9 

io.6i3562 

\l 

42 

374452 

8-63 

987465 

•  51 

388084 

9 

14 

6i3oi3 

43 

374970 

8-62 

9H7434 

.51 

9 

i3 

612464 

\l 

44 

375487 

8-61 

987403 

.52 

9 

12 

61 1916 
6ii369 

45 

376003 

8-60 

987372 

.52 

388631 

9 

II 

i5 

46 

376519 

8-5? 

987341 

.52 

3891-78 

9 

10 

610822 

14 

49 

37703d 

987310 

.52 

389724 

9 

^ 
U 

610276*  i3 

IS 

8.57 

987270 
987248 

.52 
.52 

390270 
390815 

I 

609730 
609185 

12 
II 

5o 

9.379089 

8-54 

987217 

.52 

391360 
9.391903 

9 

608640 

ry 

5i 

8.53 

9.987186 

.52 

9 

o5 

10-608097 

I 

5a 

379601 
38aM3 

852 

987,55 

.52 

392447 

9 

04 

607533 

53 

8.5i 

9«7I24 

.52 

392989 

9 

o3 

607011 
6.36469 

I 

54 

380624 

8-5o 

987092 

•52 

393331 

9 

02 

55 

38n34 

8.45 
8-48 

987061 

.52 

394073 

9 

01 

605927 
6o5386 

5 

56 

38i643 

987030 

.52 

394614 

t 

00 

4 

U 

382152 

8.47 
8-46 

986998 

.52 

395.54 

90 

604846 

3 

382661 

986467 

.52 

3^233 

8 

98 

6o43o6 

2 

59 

383i68 

8-45 

986930 

.52 

8 

97 

603767 

I 

fe^ 

383675 

8.44 

gHbtfo^ 

•52 

396771 

8 

96 

6o3239 

0 

« , 

Cofine 

J). 

ISijie 

763 

Cotang. 

1>. 

T.-^i^ 

J^:. 

2r. 


52 

(14  DBGREES.)   A 

TABLE  OF  LOGARITHMIC 

o 

Sine 

D. 

CoHiiie   D.  1  Taii^. 

5. 

Cotang. 

60 

9-383675 

8.44 

9-9869041  -52 

9.396771 

8.96 

10.603229 

I 

384182 

S.43 

986873  -53 

3973og 

8.96 

602691 

It 

a 

384687 

8.42 

98684.  -53 

397846J  8.95 

602154 

3 

385192 

8-41 

986809  -53 

398383 

8.94 

601617 

57 

4 

385697 

8.40 

986778:  -53 

3989.9 

1  8-93 

60 1 08 1 

56 

5 

386201 

8^3^ 

986746  -53 

3994551  8.92 

6oo545 

55 

6 

386704 

9867 14I  -53 

399990 

8-91 

600010 

54 

1 

387207 

8.37 

986683,  -53;   4oo524 

S.90 

1  tk 

599476 
1   598942 

53 

£ 

387709 

8-36 

98665 1 1  -53 

4o.o58 

52 

9 

!   388210 

8-35 

986619!  -53 

401591 

598409 

597876 

5i 

10 

'   3887.1 

'  8-34 

986587;  -53 

402.24 

8-87 

5o 

II 

9-389211 

8-33 

9.9865551  -53 

9-402656 

8.86 

10.597344 

8 

12 

3897.1 

8-32 

986523  -53 

403.87 

8.85 

596813 

i3 

3902.0 

8.3i 

986491  -53 

4037.8 

8.84 

596282 

tl 

14 

390708 

8.3o 

986459 

.53 

404249 
404778 

8.83 

595751 

i5 

39.206 

8.28 

986427 

.53 

8.82 

595222 

45 

i6 

39.703 

8.27 

986395 

•53;   4o53o8 

8.81 

594692 

44 

n 

392199 

8.26 

986363 

■54 

405836 

8.80 

594164 

43 

iS 

392695 

8-25 

,      986331 

•54 

406364 

8.70 
8.78 

593636 

42 

19 

39819. 

8.24 

986299:  -54 

406892 

593.08 

41 

20 

393685 

8-23 

986266 

•54 

407419 

8-77 
8.76 

592581 

^° 

21 

9-394.79 
394673 

8-22 

9.986234 

•54]  9.407945 

10-592055 

t^ 

22 

8-21 

986202 

•54 

408471 

8.75 

Hfj,^ 

23 

395.66 

8.20 

986169 

.54 

408997 
40952 1 

8.74 

37 

24 

395658 

8.19 

8.i8 

986.37 

If 

8-74 

590479 
589953 

36 

25 

396.50 

986104 

•54 

4.0045 

8.73 

35 

i6 

396641 

8.17 

986072 

•54 

410569 

8.72 

589431 

34 

^J 

397.32 

8.17 
8.16 

986039 

•54 

41.092 

8.71 

588908 

33 

397621 

986007 

•54 

4ii6i5 

8.70 

588385 

32 

?9 

398  m 

8.i5 

985974 

•54 

412.37 

8.69 
8-68 

587863 

3i 

3o 

398600 

8.14 

985942 

.54 

4.2658 

587342 

3o 

3i 

9-399088 

8.i3 

9.985909 

•  55 

9-413.79 

8.67 

10.586821 

11 

32 

399575 

8-12 

985876 

.55 

413699 

8.66 

586301 

33 

400062 

8-11 

985843 

.55 

414219 
414738 

8-65 

585781 

27 

34 

400549 
401035 

8.10 

9858.1 

.55 

8-64 

585262 

26 

35 

8.09 
8.08 

985778 

.55 

415257 
415775 

8-64 

584743 

25 

36 

40.520 

985745 

.55 

8.63 

584225 

24 

37 

4o2oo5 

8-07 

985712 

.55 

416293 

8.62 

583707 

23 

38 

402489 

8-06 

985679 

.55 

4168.0 

8.61 

583.90 

22 

39 

402972 

8-05 

985646 

.55 

417326 

8.60 

582674 

21 

40 

403455 

8-04 

9856.3 

.55 

417842 

8.59 
8.58 

582158 

20 

41 

9-403938 

8-03 

9.985580 

.55 

9.418358 

10.581642 

\t 

42 

404420 

8-02 

985547 

.55 

418873 

8.57 

581.27 

43 

404901 
4o5382 

8-01 

9855.4 

.55 

419387 

8.56 

5806.3 

n 

44 

8.00 

985480 

.55 

419901 

8.55 

580099  16 
579585 1  1 5 

45 

4o5862 

7-9^ 

985447 

.55 

420415 

8.55 

46 

406341 

985414 

.56 

420927 

8.54 

570073:  14 
578560I  1 3 

ii 

406820 

7-97 
7.96 

985380 

•  56 

421440 

8.53 

407299 

985347 

•  56 

421952 

8-52 

578048  12 

49 

407777 

7-95 

9853.4  -S^ 

42  2463 

8.5i 

577537 

" 

5o 

408254 

7-94 

985280:  .561   422974 

8.5o 

577026 

TO  ! 

;)i 

9  408731 

7-94 

9.9852471  -56 

9.423484 

8.49 
8.48 

10.576516 

I 

55 

409207 

7-93 

9852.3  -56 

424503 

576007 

53 

409682 

7.92 

985.80:  -56 

8.48 

575497 

1 

6 

54 

410.57 

7.91 

985.46:  -56 

425oii 

8.47 

574989 

55 

4io632 

985.13  -56 

4255.9 

8.46 

574481 

5 

56 

4 1 II 06 

985079!  -56 
98504D  -56 

426027 

8.45 

573973 

4 

U 

411579 

426534 

8-44 

573466 

3 

4I2052 

7-87 

98001 1  -56 

427041 

8.43 

572950 
572433 

a 

59 

412524 

•^.86 

984978  -56 

427547 

8.43 

I 

_6o_ 

412996 

7-85 

984944' 

56i 

428052 

8.42 

571948  0  1 

L 

Cosine  1 

D 

Sine   1 

T50| 

Cotang. 

D. 

Tnne.   M.  | 

sixes  AND  TAXGEMS. 

(15  DEGREES. 

)       33 

[M. 

Siiio 

D. 

Cosine 

1>. 

Tailg. 

D. 

CotAng 

o 

7-65 

9-984944 

Q -428052 

8-42 

10-571948 

60 

1 

7-84 

984910 

•371  '  428557 

8-41 

571443 

i? 

3 

7-83 

984876 

•37   429062 

8-40 

570938 

3 

414408 

7.33 

984842 

•57   429566 

8^3? 

570434 

u 

4 

^l^H-jS 

7-82 

984H08 

•57   430070 

569930 

5 

41334] 
4i58iD 

7.81 

984774 

•57   430573 

8-38 

5^^^? 

55 

6 

7-8o 

984740 

•57   431075 
•57   43.577 

fM 

54 

I 

9 

4i6a83 

984706 

563423 

53 

416751 
417217 

984637 
9846o3 

•57   4320-79 
•57i   432580 

8.35 
8-34 

567921 
567420 

l\ 

10 

417684 
<;.4iBi5o 

•57I   433o8o 

8.33 

566920 

5o 

II 

7-75 

9.984569 

•57!  9-433580 

8-32 

.0-566420 

% 

la 

4186.5 

7-74 

98433d 

•^7 

434080 

8.32 

565920 

i3 

419079 

7-73 

984500 

'V 

f^m 

8.3, 

565421 

47 

14 

419544 

7-73 

984466 

:tl 

8-30 

564922 

^i 

|5 

420007 

7-72 

984432 

435576 

8-29 
8-28 

564424 

45 

i6 

420470 

7-7» 

984397 

•  58 

436073 

563927 

44 

\l 

42qq33 
421395 

984363 

•  58 

436570 

8-28 

563430 

43 

984328 

•  58 

iP^li 

8-27 

562933 

42 

'9 

421857 

984294 

•  58 

8-26 

562437 

41 

20 

4223id 

7-67 

984239 

•  58 

438o59 

8-25 

56.941 

40 

21 

9-422778 

7.67 

9-984224 

•58!  Q-438554 

8-24 

10-56.446 

It 

32 

423238 

7.66 

984 • 90 

•  58 

439048 

8-23 

560932 

23 

42^697 

7-65 

984133 

•  58 

439543 

8-23 

560457 

u 

24 

424106 

7-64 

984120 

•  58 

440036 

8-22 

559964 

25 

424615 

7-63 

984083 

•  58 

440529 

8-21 

5%78 
558486 

35 

26 

425073 

7.62 

984050 

•  58 

441022 

8-20 

34 

U 

425530 

7-6i 

984015 

•  58 

44i5i4 

8..9 

33 

425987 

7.60 

983981 

.58 

442006 

8-19 
8-. 8 

557994 

32 

29 

426443 

7-60 

983946 

•  58 

442497 
442988 

557303 

3i 

3o 

9-427354 

7.59 

98391 1 

•  58 

8..7 

5570.2 

3o 

3i 

7-58 

9-983873 

•  58 

Q-44347q 

8-16 

10-556521 

ll 

32 

428263 

7-57 

983840 

•59|  ■  443968 

8.16 

556o32 

33 

7-56 

9838o5 

•59|   444458 

8-15 

555542 

ll 

25 

34 
35 

428717 
429170 

7.55 
7-54 

$l]lt 

•591   444947 
•59   445435 

8.14 
8..3 

555o53 
554565 

36 

429623 

7-53 

983700 

•59   445923 
•59   446411 

8-. 2 

554077 
5535B9 

24 

ll 

430075 

7-52 

983664 

8-. 2 

23 

43o527 

7-52 

983629 

•59   446898 

8-II 

553.02 

22 

39 

430978 

7-5i 

9«^?9i 

•39   447384 

8-.0 

5526.6 

21 

40 

431429 

7-5o 

983338 

•59   44.870 

8^09 

552 1 3o 

20 

41 

9431879 

7-49 

9-983523 

•59  9-448356 

8-09 
8-08 

10-55.644 

:? 

4: 

432329 

7-49 

983487 

•^   448841 

55.. 59 

43 

432778 

7-48 

983452 

•59   449326 

8-07 

550674 

n 

44 

433226 

7-47 

983416 

•59   449S10 

806 

550.90 

.6 

45 

433675 

7-46 

983381 

•59   450294 

8-06 

§49706 

.5 

46 

434122 

7-45 

983345 

•39:   450777 

8-o5 

549223 

548740 

14 

% 

434569 

7-44 

9832i8 

•59,   45.260 

8-04 

i3 

435016 

7-44 

•60,   45.743 

8-03 

548937 

12 

49 

435462 

7-43 

•60    452225 

8-02 

547775  II  1 

5c 

435908 
9.436353 

7-42 

983202 

•60!  452706 

8-02 

547294 

10  1 

5i 

7-41 

9-983166 

•60;  9.453.87 

8-01 

10-5468.3 

t 

5a 

436798 

7  40 

983 i3o 

•6oj   453668 

8-00 

54633: 

53 

437242 

7-40 

983094 

•<>0|   454148 

7-99 

545852 

54 

437686 

?:^ 

983038 

•60   454628 

?:^ 

545373 

55 

4.^8129 

983022 

•60   455.07 

54489.' 

56 

43'5572 

7-37 

982986 

•60   455586 

]'-U 

544414 

U 

439014 

7-36 

983950 

•601   456064 

543936 

439456 

7-36 

982914 

•60   456542 

7-96 

543458 

^ 

'^l 

7.35 
7-34 

982878 
982842 

•6o|   457019 
•60   457496 

7-95 

7-94 

D. 

542981 
54a5o4 

0 

1 

Cosiue 

1). 

_  S^iue  _ 

74''l  Cotaug. 

84 

(16 

DEGRKE8.)   A  TABLE  OP  LOGARITHMIO 

M. 

0 

Sine 

L\ 

Cosine 

1). 

Taii^. 

D. 

Cotansr.  1 

9-440338 

7-34 

9-982842 

.60 

9.457496 

7-94 

10- 542504'  60 

I 

440778 

-33 

982805 

.60 

457973 

7-93 

5420271  59 
54i55r'  58 

3 

441218 

-32 

982769 

.61 

458449 

7.93 

3 

441658 

-31 

982733 

.61 

458923 

7.92 

541075  57 

4 

442096 

.31 

982696 

•61 

459400 

7.91 

540600  56 

5 

442535 

7 

•3o 

982660 

.61 

459875 

7-90 

540. -.5  55 

6 

442973 

7 

-.11 

982624 

•61 

460349 
460823 

539651  34 

I 

443410 

- 
/ 

982587 

•61 

539177  53 
538703  53 

443847 

7 

•27 

9825511 

-61 

461297 

9 

444284 

- 
/ 

•27 

982514 

-61 

461770 

j'ss 

538230  5i 

10 

444720 

•26 

982477 

•61 

462242 

7-87 

537758,  5o 

II 

9 -445 1 55 

7 

•25 

0- 982441 

.61 

9-462714 

7.86 

.'3-537286:  49 
536814'  48 

12 

445590 

•24 

982404 

.61 

463 186 

7-85 

i3 

446025 

7 

•23 

982367 

•61 

463658 

7.85 

5:6342  47 

14 

446459 

•23 

982331 

-61 

464129 

7-84 

535871'  46 

i5 

446893 

22 

9822941 

•61 

464599 

7-83 

535401!  45 

i6 

.  447326 

21 

982257 

•61 

465069 

7-83 

53493 r  44 

*l 

447759 

^ 

20 

982220 

-62 

465539 

7.82 

534461 i  43 

i8 

448191 

1 

20 

982183 

.62 

466008 

7.8. 

533992!  42 

^9 

448623 

' 

Is 

982146 

.62 

466476 

7.80 

533524  41 

20 

449054 

982109 

•62 

466945 

7.80 

533o55i  40 

21 

9-449485 

7 

17 

9-982072 

.62 

9-467413 

7-79 

10-5325871  39 
532120!  38 

22 

449915 

16 

982035 

-62!   467880 

7.78 

23 

45o345 

16 

981998 

.621   468347 

7.78 

53 1 653 i  37 

24 

450775 

i5 

981961 

-62 

468814 

I'll 

53 r 1 861  36 

25 

45 1204 

14 

981924 

-62 

469280 

7.76 

530720  35 

26 

45i632 

i3 

981886 

•62 

469746 

7-75 

530254!  34 

11 

452060 

7 

i3 

981849 

•62 

4702 1 1 

7-75 

529780;  33 

452488 

_ 

12 

981812 

-62 

470676 

7-74 

529324I  32 

29 

452915 
453342 

1 1 

981774 

•62 

471141 

7-73 

528859!  3 1 
528395  3o 

3o 

7 

10 

981737 

-62 

471605 

7-73 

3i 

9-453768 

10 

9-981699 

•63 

9-472068 

7-72 

10.527932  2q 

32 

454194 

:i 

981662 

-63 

472532 

7-71 

527468|  28 

33 

454619 

981625 

•63 

472995 
473457 

7.71 

527005I  27 

34 

455o44 

07 

981587 

-63 

7.70 

526543 

26 

35 

455469 
455893 

07 

981549 

•63 

473919 

7.69 

526081 

25 

36 

06 

981512 

-63 

474381 

-tl 

5256 I Q I  24  j 

ll 

4563 16 

o5 

98' 474 

•63 

474842 

525i58 

23 

456739 

04 

981436 

•63 

4753o3 

7-67 

524697 

22 

39 

457162 

04 

981399 

•63 

475763 

7-W 

524237 

21 

40 

457584 

o3 

981361! 

•63 

476223 

7.66 

523777 

20 

41 

9 -458006 

02 

9-981323 

•63 

9-476683 

7-65 

10.523317 

522858 

\t 

42 

458427 

01 

981285 

•63 

477142 

7.65 

43 

458848 

01 

981247 

-63 

477601 

7-64 

52  2399 

]l 

44 

459268 

CO 

981209 

•63 

478059 

7-63 

521941 

45 

459688 

9^^ 

981171 

•  63 

478517 
478975 

7-63 

521483 

i5 

46 

460108 

981 i33 

-64 

7.62 

52I025 

14 

il 

460527 

98 

981095 
981057 

-64 

479432 

7.61 

520568 

i3 

46?364 

97 

.64 

i& 

7.61 

520I11 

12 

^9 

96 

981019 

.64 

7.60 

519655  II 

5o 

461782 

95 

980981 

•  64 

480801 

7.59 

519199  10 

10.518743'  9 

518288;  8 

5i 

5-462199 

95 

9-980942 

.64 

9-481257 

]i^ 

52 

462616 

94 

980904 
980866 

•  64 

481712 

33 

463o32 

93 

•  64 

482167 

I'^i 

fi7833   7 
5,7379  6 

516925I   3 

54 

463448 

93 

980827 

.64 

482621 

1-^1 

55 

463864 

92 

980789 

•64 

483075 

7-56 

56 

464279 

91 

980750 

•  64 

483529 

7.55 

516471J  4 

ll 

464694 

90 

980712 

•  64 

483982 

7.55 

5i6oi8i  3  1 

465 I 08 

To 

980673 

-64 

484435 

7-54 

5i5565'  2  , 

^ 

465522 

980635 

-64 

484887 

7.53 

5i5ii3i   I  ; 

6o 

4b5935 

6- 

88 

980596 

-64 

485339 

7.53 

5i466i 

"  ; 

Coeino 

D.   J 

Sine   -J 

30 

(Jotang. 

D. 

Taiisr. 

JLJ 

SIVK8  AND  TANGENTS. 

(17  DEORKE8. 

) 

35 

BT.  1   Sino  1 

D. 

Cosiiio 

D. 

Tan?. 

D. 

Cotancr. 
10.514661 

! 

0 

9-465935 
466348 

6-88 

Q. 980596 

.64 

9.485339 

7.55 

60 

I 

6-88 

^  98o5d8 

.64 

485791 

7.5a 

514209 
5i3758 

u 

a 

466761 

6.87 
6-86 

980519 

•  65 

486242 

7.51 

3 

467173 
467585 

9S0480 

.65 

486693 

7.51 

5i33o7 

u 

4 

6-85 

980442 

.65 

487143 

7.50 

512857 

56 

5 

467996 

b-St 

980403 

•  65 

487593 

7-49 

5 12407 

^^ 

6 

468407 

6-84 

980364 

.65 

488043 

lit 

iwtu 

li 

I 

468817 

6-83 

980325 

.65 

488492 

53 

4S9227 

6-83 

980286 

.65 

488941 
489390 
489838 

7-47 

5iio59 

32 

9 

10 

469637 
470046 

6-82 
6.81 

980247 
98020S 

•  65 

•  65 

]■■% 

5io6io  31 
510162  30 

11 

9-470455 

6.80 

9-980169 

.65 

9-490286 

7.46 

10.509714!  40 
509267  48 

u 

470863 

6.80 

980180 

.65 

490733 

7.45 

i3 

47127' 

Ui 

980091 

•  65 

491180 

7-44 

508820 

47 

14 

471679 
472086 

980002 

•  65 

491627 

7-44 

5o8373 

46 

i5 

6-78 

980012 

•65 

492073 

7-43 

507927 

45 

i6 

472492 

6-77 

979973 

•65 

492019 

7-43 

507481 

44 

\l 

472898 

6-76 

979934 
979893 

.66 

492963 

7-42 

507035 

43 

473304 

6.76 

.66 

493410 

7-41 

5o65qo 

42 

'9 

473710 

6.75 

979835 

.66 

493854 

7.40 

5o6i46 

41 

20 

4741 1 5 

6-74 

979816 

.66 

494299 
9-49474i 

7.40 

503701 

40 

21 

9-474510 

474923 
475327 

6.74 

9.979776 

.66 

7.40 

10-503237 

It 

22 

6.73 

979737 

.66 

493186 

vu 

504814 

38 

23 

6-72 

979697 

.66 

495630 

504370 

37 

24 

475730 

6.72 

979638 

.66 

496073 

7-37 

503927 

36 

25 

476133 

6.71 

979618 

.66 

4965 1 5 

7-37 

5o3485 

35 

26 

'2 

476536 
476938 
477340 

6-70 
6-69 

979579 
979539 

.66 
.66 

ti^ 

7.36 
7.36 

5o3o43 
5o26oi 

34 
33 

2S 

6-69 
6-68 

979499 

•  66 

497»4i 

7.35 

5o2 1 39 
501718 

32 

?9 

477741 
478142 

979439 

.66 

498282 

7.34 

3i 

3o 

6-67 

979420 

.66 

498722 

7-34 

501278 

3o 

3i 

9-478542 

6.67 
6-66 
6-65 

9-979380 

.66 

9-499'63 

7.33 

io.5oo837 

11 

27 

32 

33 

478942 
479^42 

979340 
979300 

.66 
-67 

499603 
5ooo42 

7.33 
7-32 

499938 

34 

479741 
4^0 I 40 
480539 

6-65 

979260 

-67 

5oo48i 

7-31 

4993 '9 

26 

35 

36 

6-64 
6.63 

979220 
979180 

-67 
.67 

500920 
5oi359 

7-31 
7-3o 

499080 
498641 

25 

24 

u 

480037 
481334 

6-63 

979 « 40 

-67 

mm 

7.30 

498203 

23 

6-62 

979100 

•67 

]:lt 

497765 

22 

39 

481731 

661 

979059 

.67 

502672 

497328 

21 

40 

482128 

6.61 

9790 I Q 
9 •978979 

-67 

5o3io9 

7.28 

496891 

20 

41 

9-482525 

6.60 

.67 

9  503546 

7-27 

10-496434 

'5> 

42 

482921 
4833 1 6 

6-59 

978o3o 
978898 

.67   503982 

7.27 

496018  la 

43 

6-5? 

.671   5o44iB 

7.26 

493582  17 
493146  16 

44 

483712 

978838 

.67,   504854 

7-25 

45 

484107 

6.57 

978817 

.67!   505289 

7-25 

49471 1  '5 

46 

484501 
484895 
485289 

6.57 
6-56 

978777 
978736 

.67I   5o5724 
.67!   5o6i59 
•68!   506593 

7-24 
7-24 

494276  14 
493841  «3 

655 

978696 

7-23 

493407'  12 

f^ 

485682 

6. 55 

978635 

•68   507027 

7-22 

492973  " 

5o 

486075 

6.54 

978615 

.68   507460 

7.72 

492340  10 

Si 

9-486467 

6-53 

9-978574 

•68;  9.507893 

7-21 

ic  492107:  9 
491674  " 

53 

486860 

6-53 

978533 

.68 

5o8326 

7-21 

53 

487251 

6.5a 

978493 

•  68 

508759 

7.20 

491241;  7 

54 

487643 
488034 

6.5i 

978432 

.68 

509191 

7. 19 

490809  6 
400378,  5 

55 

6.5i 

978411 

.68 

509622 

?::? 

56 

488424 

6.5o 

978370 

.68 

5ioo54 

1] 
5q 
6o 

488814 
489204 

6.5o 
649 
6.48 
6.48 

978247 

978206 

.68 
.68 
.68 
.68 

5 1 0485 
510916 
5ii346 
511776 

7.18 
,..6 

48^5   ■ 
489C84 
488654 
488234 

3 

2 
1 
0 

Conine 

!   D. 

Sine 

7  2- 

Cotaiig. 

D. 

Taiij^r. 

.?•  i 

30 

(18  DEGREES.)   A 

TABLE  OF  LOCiARlTHMlC 

Ti: 

Sine 

D. 

Cosine  1  1). 

Taiicr. 

1   ^• 

Cotaug. 

0 

9-489082 

6-48 

9.978206'  .68 

9.511776 

1  7-i6 

10.488224  60 

I 

490071 

1  6-48 

978165;  -68 

5l2206 

7-16 

4877941  59 

7 

4907^9 

!  6-47 

978124'  -68 

512635 

7.15 

487363 

58 

3 

49n47 

!  6-46 

978083  -69 

5 1 3064 

7-14 

486936 

57 

4 

491535 

'  6.46 

978042;  -69 

513493 

7-14 

486307 

56 

5 

491922 

6.45 

978001  -69 

51392, 

7-i3 

486079 

55 

6 

492308 

i  6-44 

977950  -69 
9770,8  .69 
977077:  -69 

514349 

7-i3 

485651 

54 

7 

492695 
493081 

6-44 

514777 

7-12 

485223 

53 

8 

^.43 

5i52oi 

7-12 

484796 

5i 

9 

493466 

6.42 

977835  .69 

5i563, 

7-II 

484369 
483943 

5, 

IC 

49385 I 

6-4J2 

977794  -69 

5i6o57 

7-10 

5d 

11 

9 -454:35 

6.4! 

9-9777321  -69 

9.5,6484 

7.10 

10.48J316 

1? 

12 

494621 

6.41 

977711  -69 

5169I0 
517335 

7-09 

483090 

i3 

495co5 

6.40 

977660'  -69 
977628^  -69 

7-0? 

482665 

47 

14 

495388 

6.39 

51776, 
5i§i85 

482239 
4818,5 

46 

i5 

493772 

tl^ 

977586 

:   .69 

7-o8 

45 

i6 

496154 

977544 

1  -70 

5i86io 

]:2 

48,390 

44 

n 

496537 

6.37 

9775o3 

!  .70 

5,9034 

480966 
480342 

43 

i8 

496919 
497 JO  I 

6-37 

977461 

i  -70 

519458 

7-o6 

42 

19 

6-36 

977419 

.  .70 

5,9882 

7-05 

480118 

41 

20 

497682 

6-36 

977377 

.70 

52o3o5 

7-o5 

479695 

40 

21 

9-498064 

6-35 

9.977335 

•70 

9.520728 

7-04 

10.479272 

39 

38 

22 

498444 

6.34 

977293 
977231 

•70 

521, 5t 

7-03 

478849 

23 

498825 

6.34 

•70 

52,573 

7-03 

478427 

ll 

14 

499204 

6.33 

977209 

•70 

521995 

7.03 

478003 

i5 

499584 

6.32 

977167 

•70 

522417 

7.02 

477583 

35 

i6 

499963 

6-32 

977125 

•70 

522838 

7-02 

477162 

34 

n 

5oo342 

6.3i 

977083 

•70 

52325g 

7-01 

476741 

33 

i8 

500721 

6-31 

977041 

•70 

52368o 

7-0, 

47^)320 

32  1 

i*^ 

501099 

6.3o 

976999 

•70 

524100 

7-00 

475900  3i  1 

io 

501476 

6-29 

976937 

•70 

524520 

6.99 

475480 

3o 

ti 

9-5oi854 

6-29 
6.28 

9.976014 
976872 

•70 

9-524939 

6-99 

,0.475061 

29 

I2 

50223l 

•71 

525359 

6.9^ 

474641 

28 

53 

502607 

6.28 

97683o 

•71 

525778 

6.98 

474222  27 

H 

502984 

6.27 

976787 
976745 

•71 

526197 

6-97 

473803'  26 

)5 

5o336o 

6-26 

•71 

526615 

6-97 

473385  25 

16 

503735 

6.26 

976702 

•71 

527033 

6.96 

472967  24 

!^ 

5o4iio 

6-25 

976660 

•7' 

52745, 

6-96 

472349  23 

504485 

6-25 

976617 

•71 

527868 

6.95 

472132  i2 

^9 

504860 

6-24 

976574 

•71 

528285 

6-95 

47i7'5,  2, 

40 

5o5234 

6-23 

976532 

•71 

528702 

6.94 

471298  20 

4i 

9 -505608 

6-23 

9.976489 

•71 

9-5291,9 

6.93 

,0.47088,  IQ 

i2 

505981 
5o6354 

6-22 

976446 

•71 

529535 

6.93 

470465  18 

43 

6-22 

976404 

•71 

529930 

6.93 

47oo5o|  17 

44 

506727 

6-21 

976361 

•71 

53o366 

6.92 

469634'  16 

45 

507099 

6-20 

676318 

•7' 

530781 1 

6.9, 

469219'  ,5 
468804;  4 

46 

507471 

6-20 

976275 

•71 

53,196 

6.91 

47 

507843 
508214 

6-19 

976232 

•72 

53i6ii; 

6.90 

468389 

i3  i 

48 

6-19 
6-i8 

976189 

•72 

532025 

6-90 
6.89 

467973 

12  I 

49 

5o8585 

976146 

•72 

532439 

467361 

"  1 

5o 

508956 
9.509326 

6.18  1 

976103 

•72 

532853 

tli 

467147 

ID  1 

5i 

6-17  : 

9  976060 

.72 

9 '533266 

10.466734 

9  I 

5s 

5,w^; 

6.16  1 

976017 

•72 

533679 

6.88 

466321   8  1 

53 

:jioo65 

6.16  i 

975974 

•  72 

534092 

6.87 

465908   7 
465496  6 
465o84i  5 

54 

510434! 

6-15 

975930 

97D887 

•72 

534504 

6.87 

55 

5io8o3 

6.i5 

•72I 

534916 

6.86 

56 

511172 

6-14  1 

975844  -72 

535328 

6-86 

464(372   4 

U 

5ii54o 

6.i3  ! 

975800!  .72 

535739' 

6-85 

464261   3 

511907 

6-13 

975757I  .72 

536i5o 

6.85 

463850   2 

59 

512275 

6-12 

973714!  -72 

536561 ; 

6-84 

463439'  I 

(jo 

512642 

6-12 

9736701  .72 

536972 

6-84 

463028, 

0 

Coaiiio 

__.I^-.__ 

Siijo  171° 

Cotaiig.^l 

_i>. 

_Ttiiig^l 

M^ 

SINES  AND  TANQENTfi. 

(19  L»EGUEEri. 

1 

8^ 

o 

Sine 

D. 

Cosiiio 

D. 

_Tuug^ 

1  ^' 

Cotang. 

._ 

0-512642 

6-12 

9.975670 

•73 

9-536972 

6-84 

10-463028;  60 

I 

5i3oo9 

61. 

i^m 

•73 

537382 

6-83 

462618 

!? 

2 

5i3373 

6  I. 

•73 

537792 
538102 
53861 1 

6.83 

462208 

3 
4 

51374. 
514107 

6-10 
609 

975539 
975496 
973432 

•73 
•73 

6.82 
6.82 

461798 
46.389 

U 

5 

514472 

6.09 
6.08 

.73,   539020 

6.8( 

460980 
460571 

55 

6 

5.4837 

975408 

•73 

539429 

6.81 

54 

I 

5.5202 

6.08 

975365 
975321 

•73 

539837 

6-80 

460163 

53 

5i5566 

6-07 

•73 

540243 

6-8o« 

459755 

52 

9 

10 

5.5930 

5.6204 

9.516657 

tS 

Q. 975189 

.73;   540653 
.73'   54.061 

6-79 
6.70 
6. 78 

45q347 

438939 

10-458532 

5i 

5a 

if 

6.o5 

•73 

9-54.468 

tt 

13 

5.7020 

6-05 

975.43 

•73 

54.875 

6-78 

458.25 

i3 

5.7382 

5-04 

975.0. 

•73 

542281 

6-77 

457719 

ii 

45 

14 
i5 

5.2745 
518107 
5.8468 

604 
6  03 

975057 

975oi3 

•73 
•73 

542688 
543094 

6.76 

4573  m 
456906 
456501 

i6 

6-o3 

974960 
974925 

•74 

543499 

6.76 

44 

'.I 

5.8829 

6-02 

•74 

54390! 

6.75 

456095 

43 

519190 

6-0. 

974880 

•74 

5443.0 

6.75 

455690 
455285 

42 

'9 

5195DI 

6-0. 

974836 

•74 

544715 

6.74 

4. 

20 

519911 

600 

974792 

•74 

545119 

6-74 

454881 

40 

21 

9-52027. 

600 

9-974748 

•74 

9-545524 

6.73 

10-454476 

^ 

22 

^   52063. 

5-99 

974703 

•74 

545928 
546331 

6.73 

454072 

23 

520990 

5-99 

974659 

•74 

6.72 

453669 
453265 

u 

24 

521349 

5-98 

974614 

•74 

546735 

6-72 

25 

521707 

5-98 

974570 

•74 

547.38 

6.7. 

452S62 

35 

26 

522066 

5.97 

974525 

•74 

547540 

6-71 

452460 

34 

11 

5224^4 

5.96 

974481 

•74 

liit$ 

6.70 

452057 

33 

5227S1 

5.96 

974436 

•74 

6-70 

431655 

32 

29 

523i3S 

5-95 

97439. 

•74 

548747 

6-69 

451253 

3i 

3o 

523495 

5.95 

974347 

•75 

549149 

6-69 
6-68 

45o85i 

3o 

3i 

9-523832 

5.94 

9.974302 

•75 

9.'j4955o 

io-45o45o 

It 

32 

524208 

5.94 

974257 

•75 

P^[ 

6-68 

450049 
449648 

33 

524564 

5.93 

9742.2 

•75 

6-67 

'\ 

34 

524920 

5.93 

974.67 

•75 

550752 

6-67 

449248 
448848 

35 

523275 

5.92 

974122 

•75 

55ii52 

6-66 

25 

36 

525630 

5-9. 

974077 

•75 

55i552 

6-66 

448448 

24 

^2 

525584 

5.91 

974032 

.75 

55.952 
552351 

6-65 

448048 

23 

526339 
526693 

5.90 

973987 

•75 

6-65 

447649 

22 

39 

5% 

973942 

•75 

552750 

6-65 

447250 

21 

40 

527046 

973^^7 

•75 

553149 
9-553548 

6-64 

44685. 

20 

41 

9.527400 

5-8q 

9-973832 

•75 

6-64 

10.446452 

\t 

43 

522753 
528 1 o5 

5.88 

973307 

•75 

553946 
534344 

6-63 

446054 

43 

5-88 

973761 

•75 

6-63 

445656 

\l 

44 

528458 

5.87 

973671 

•76 

55474. 

6.62 

445259 

45 

528810 

It 

•76 

555 1 39 
555536 

6-62 

444861 

.5 

46 

529161 

973625 

.76 

6-61 

444464 

.4 

s 

529513 

5-86 

9735SO 

•76 

555933 
556J29 

6-6. 

444067 

i3 

529864 

5-85 

973535 

•76 

6-60 

443671;  n   1 

49 

53021 5 

5.85 

?73489 

•76 

55672! 

660 

4432751  I 

5o 

53o565 

5.84 

973444 

•76 

55712. 

6.59 

442870  1.) 

.0-442483:  0 

442087'  8 

5i 

o.53o9i5 

5.84 

9.973398 

.76 

9.557517 

6.59 

5a 

53.265 

5-83 

973352 

.76 

557913 
5583o8 

tu 

53 

53i6.4 

5.82 

973307 

.76 

441692J  7 
441298I  6 

54 

531963 
532312 

5.83 

973261 

.76 

558702 

6.58 

55 

5.81 

9732.5 

.76 

559097 

6.57 

4409031  5  , 

56 

532661 

5.81 

973169 

.76 

5598^5 

tu 

440509 
4401.5 

4 

59 

533009 

5.80 

973.24 

•76 

3 

533357 
533704 

5. So 

973078 
973o32 

.76 
•77 

56io66 

6  56 
6. 55 

439721 

2 

I 

60 

534052 

972986 

•77 

6.55 

43?934 

0 

Coaiud 

I). 

Sine 

TOO 

Cotang. 

1> 

TaiM?. 

88 


(20    DEGREES.)       A    TABIE    OIT    LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

|I>. 

1  Tang. 

D. 

1  Cotancr. 

n 

o 

9.534052 

5.78 

9.972986 

•77 

9.56106b 

6.55 

10.438934'  60 

I 

534399 

5-77 

972940 

•77 

561459 

6.54 

438341!  59 
438.49  58 

2 

D34743 

5-77 

972894 

•77 

56i85i 

6.54 

3 

535092 
53543H 

t^ 

972848 

•77 

562244 

6.53 

^  u 

4 

972802 

•77 

562636 

6.53 

5 

535783 

5.76 

972755 

•77 

563028 

6.53 

436972  55 

6 

536129 

5.75 

972709 
972663 

•77 

563419 

6.52 

436381  54 

I 

536474 

5-74 

•77 

563811 

6.52 

436189  53 
435798  52 

5368 1 8 

5-74 

972617 

•77 

564202 

6.5i 

9 

537.63 

5.73 

972570 

•77 

564592 

6.5i 

435408  5i 

10 

537307 

5.73 

972524 

•77 

564983 
9-565373 

6.5o 

435017  5o 

11 

9-537851 

5.72 

9.972478 

•77 

6.5o 

10.434627  49 
434237 i  48 

12 

538194 

5.72 

972431 

•78 

565763 

6-49 

i3 

538538 

5.7, 

972385 

.78 

566 1 53 

6-49 

43JS47  47 
433438  46 

14 

538880 

5.71 

972338 

.78 

566542 

6-49 
6.48 

i5 

539223 

5.70 

972291 

•78 

566932 
567320 

433068  45 

i6 

539565 

5.70 

972245 

.78 

6.48 

432680;  44 

\l 

539907 

5.69 

972198 

.78 

568486 

6-47 

432291 j  43 

540249 

t^ 

972IDI 

.78 

6-47 
6-46 

431002;  42 
43i5.4  41 

»9 

540590 

972!o5 

.78 

20 

540931 

5.68 

972o58 

.78 

568873 

6-46 

43 1.271  40 

21 

Q. 541272 

5.67 

9-972011 

.78 

9.569261 

6-45 

10.430739'  39 
43o352!  3§ 

32 

54i6i3 

5.67 

97 '964 

.78 

569648 

6-45 

23 

541953 

5-66 

971017 
971870 

.78 

570035 

6-45 

429965:  37 
429578;  36 

24 

542293 

542632 

5-66 

•78 

570422 

6.44 

25 

5-65 

971823 

•78 

57158. 

6-44 

429.9.  35 

26 

542971 

5-65 

97  J  776 

.78 

6.43 

4288o5i  34 

27 

543310 

5-64 

971729 

•79 

6.43 

428419'  33 
428033  32 

28 

543649 

5-64 

97 » 682 

•79 

571967 
572352 

6-42 

29 

544325 

5-63 

971635 

•79 

6-42 

4276481  3i 

3o 

5-63 

971588 

•79 

572738 

6-42 

427262  3o 

3i 

Q. 544663 

5-62 

9.971540 

•79 

9-573.23 

6-41 

10.426877:  2Q 
426493  28 

32 

545000 

5.62 

971493 

•79 

573507 

6.41 

33 

545338 

5.61 

971446 

•79 

573892 

6.40 

426.08  27 
425724:  26 

34 

545674 

5-6i 

971398 

•79 

574276 

6.40 

35 

54601 1 

5.60 

9713DI 

•79 

574660 

6.39 

425340  25 

36 

546347 
546683 

5.60 

97i3o3 

•79 

575044 

6.39 

4249561  24 

424573;  23 

37 

5.59 

971256 

•79 

575427 

6.39 
6.38 

38 

547019 

y? 

971208 

•79 

5738.0 

4241901  22 

39 

547354 

971161 

•79 

576.93 

6.38 

423807 1  21 

40 

547689 

5.58 

971113 

•79 

576576 

6.37 

423424!  20 

41 

9.548024 

5.57 

9-971066 

.80 

9-576958 
577341 

tu 

10.423041!  19 
422639!  18 

42 

548359 
548693 

5.57 

971018 

.80 

43 

5.56 

970970 

.80 

577723 

6-36 

4222771  17 

44 

549027 

5.56 

970022 

•  80 

578104 

6.36 

421896!  16 

45 

549360 

5.55 

970874 

.80 

578486 

6.35 

42i5i4!  i5 

46 

549693 

5-55 

970827 

.80 

578867 
579248 

6.35 

421.33  14 

ii 

550026 

5.54 

970779 
970731 

.80 

6.34 

420732'  1.3 

55o359 

5.54 

.80 

579629 

6-34 

42037 II  12 

49 

550692 

5.53 

970683 

.80 

580009! 

6.34 

41999.1  II  1 

5o 

55i024 

5-53 

970635 

.80 

53o389| 

6-33 

419611 

10 

5i 

9.55i356 

5.52 

9-970586 

.80 

9-5807691 

6.33 

10-419231 

4i885i 

? 

52 

551687 
552018 

5.52 

97o538| 

.80 

58.. 491 
58.5281 

6.32 

53 

5-52 

970490 

•  80 

6-32 

418472 

7 

54 

552349 

5.5i 

970442 

.80 

58.907! 

6-32 

4.B093;  6  1 

55 

552680 

5.5i 

970394 

.80 

5822861 

6.3. 

417714  5 

56 

553010 

5.5o 

970345 

•  81 

582665' 

6-3. 

4.7335 

4 

U 

553341 

5.50 

970297 

.81 

583043! 

6-30 

416378 

3 

553670 

5.49 

970249 

.81 

583422 

6-30 

2 

59 

554000 

5.45 
5.48 

970200 

•81 

583800 

6.29 

416200!  I  1 

60 

554329 

970152 

.81 

584177 

6- 29 

415823 

0 1 

Oofeiiie 

b. 

Sine 

69°, 

Cotang. 

1).      1 

Tang. 

M.  I 

SINKS 

AND  TANOENT8. 

(21  DKUREKS/ 

3 

M. 

o 

Si  no 

D. 

Cosine 

J>^ 

Tang. 

D. 

Cotanjf. 

9-554329 

5 

48 

9970152 

.81 

9-5a4i77 

6-29 

10 •415823  6c 

I 

554658 

5 

48 

970103 

.81 

58455D 

6 

\% 

415445  5o 
4i5o68  58 

2 

554987 
5553  ID 

5 

47 

970055 

.81 

584932 

6 

3 

5 

47 

970006 

.81 

585309 

6 

28 

414691  57 

4 

555643 

5 

46 

969957 

.81 

585686 

6 

27 

414314  56 

5 

555971 

5 

46 

969909 

•  81 

586062 

6 

27 

4139381  55 

6 

556299 

5 

45 

969860 

.81 

586439 

6 

27 

4i356i|  54 

I 

556626 

5 

45 

969811 

.81 

586813 

6 

26 

4i3i85 

53 

556953 

5 

44 

969762 

.81 

587190 

6 

26 

412810 

53 

a 

557280 
557606 

5 

44 

9697" 4 

.81 

587566 

6 

25 

412434 

5i 

r 

10 

5 

43 

969665 

.81 

587941 

6 

25 

412059 

5o 

II 

9-557932 

5 

43 

9-969616 

.82 

9-5883i6 

6 

25 

10.411684 

49 

12 

558258 

5 

43 

969567 

.82 

588691 

6 

24 

411309 

48 

i3 

558583 

5 

42 

969518 

.82 

589066 

6 

24 

410Q34 
410360 

47 

14 

558909 

5 

42 

9^^469 

.82 

589440 

6 

23 

46 

i5 

559234 

5 

41 

969420 

82 

589814 

6 

23 

410186 

45 

i6 

559558 

5 

41 

969370 

.82 

590188 

6 

23 

4098 1 2 

44 

12 

559883 

5 

40 

969321 

.82 

590062 

6 

22 

409438 

43 

560207 

5 

40 

969272 

•82 

590935 

6 

22 

409065 
408692 

42 

»9 

56o53i 

5 

^9 

969223 

.82 

59«3o8 

6 

22 

41 

20 

56o855 

5 

39 

969173 

.82 

591681 

6 

21 

408319 

40 

21 

9-561178 

5 

38 

9-09>24 

.82 

9-592054 

6 

21 

10-407946 

^9 

22 

56i5oi 

5 

38 

969075 

•82 

592426 

6 

20 

407374 

33 

23 

561824 

5 

V 

969025 

•82 

592798 

6 

20 

407202 

^"^ 

U 

562146 

5 

V. 

968976 

•82 

593170 

6 

•9 

406829 

36 

25 

562468 

5 

36 

9O8926 

•83 

593542 

6 

\l 

406458 

35 

26 

562790 

5 

36 

968877 

•83 

593914 

6 

406086 

34 

'i 

563112 

5 

36 

968827 

•83 

594285 

6 

18 

405715 

33 

563433 

5 

35 

968777 

•83 

594656 

6 

18 

405344 

32 

29 

563755 

5 

35 

968728 

•83 

595027 

6 

n 

404973 

3i 

3o 

564075 

5 

34 

968678 

•83 

595398 
9-595768 

6 

17 

404602 

3o 

3i 

9-564396 

5 

34 

9-968628 

•83 

6 

\i 

10.404232 

29 

32 

564716 

5 

33 

968578 

•83 

596138 

6 

403862 

28 

33 

565o36 

5 

ZZ 

96S528 

•83 

5q65o8 

6 

16 

403492 

27 

34 

565356 

5 

32 

968479 

•83 

596878 

6 

16 

4o3i22 

26 

35 

565676 

5 

32 

968429 

•83 

597247 

6 

i5 

402753 

25 

36 

565095 
5663 1 4 

5 

3i 

968379 

•83 

597616 

6 

i5 

402384 

24 

!2 

5 

3i 

968329 
968278 

•83 

m 

6 

i5 

40201 5 

23 

566632 

5 

3i 

•83 

6 

14 

401646 

22 

39 

56695. 

5 

3o 

968228 

•84 

598722 

6 

14 

401278 

21 

z 

567369 

5 

3o 

96S17S 

.84 

599091 

6 

i3 

10 -40054 I 

20 

41 

9-567587 

5 

29 

9-968128 

.84 

9-599459 

6 

i3 

\l 

42 

567904 

568222 

5 

% 

968078 

•84 

599827 

6 

i3 

400173 

43 

5 

96S027 

.84 

600194 

6oo562 

6 

12 

399806 

•7 

44 

568539 
568856 

5 

28 

967977 

•84 

6 

12 

399438!  16 

45 

5 

28 

967027 
967876 

•  84 

600929 

6 

II 

398704 1  «4 

46 

569172 

5 

27 

.84 

601296 
601662 

6 

II 

% 

569488 

5 

5? 

9678:6 

•  84 

6 

II 

398338,  1 3 
397971!  12 

569804 

5 

967775 

•  84 

602029 
602393 

6 

10 

i9 

570120 

5 

26 

9677:«5 

•  84 

6 

10 

397605!  II 

5o 

570435 

5 

25 

967674 

•  84 

602761 

6 

10 

397239;  10 

io^396873,  0 

3965071  8 

5i 

9-570751 

5 

25 

9-967614 

.84 

9-6o3i27 
6o34q3 

6 

09 

53 

571066 

5 

24 

967573 

•  84 

6 

09 

53 

57 1 38c 

5 

24 

9675'..2 

.85 

6o38d8 

6 

S 

396142  7 

54 

57i6q5 

5 

23 

967471 

•  85 

604223 

6 

395777  6 

S5 

V^ 

5 

23 

967421 

•85 

604588 

6 

08 

395412  5 

56 

5 

23 

967370 

.85 

604953 
6o53i7 

6 

07 

3^68] 

4 

% 

572636 

5 

22 

967310 

.85 

6 

07 

3 

572950 

5 

22 

96726^1 

•85 

6o5682 

6 

-1 

394318 

2 

59 

573263 

5 

21 

967217 

•85 

606046 

6 

"^ 

1 

60 

573575 

5-21 

967166 

•85 

606410 

6 

06 

0 

Cosine 

D. 

Sine 

«8o 

Colang. 

D. 

Taiig. 

40 

(22 

DEGKEES.)   A 

TABLK  OF  LOGARITHMIO 

u7 

Sine 

D. 

Coaiiie  |  D.  |  Tuntr. 

D. 

1  Coi&ng. 

0 

9-573575 

573888 

5.21 

9-967166,  .85  9-6o64ic 

6-06 

10-393590  60 

I 

5.20 

967115,  .85 

606773 

6-06 

3928681  58 

2 

574200 

5.20 

967064!  -85 

607137 

1  6-o5 

3 

574512 

5.19 

967013 

.85 

607500 

6-05 

392500I  57  1 

4 

574824 

5.19 

966961 

.85 

607863 

6-04 

392187 

56 

5 

573136 

5.19 
5.18 

066859 

.85 

608223 

6-04 

891770 

55 

6 

575447 

.85 

6o8588 

6-04 

391412;  54 

I 

575758 

5.18 

966808 

.85 

6o8q5o 
6093 1 2 

6-o3 

391050I  53 

576069 

5-17 

966756 

.86 

6-o3 

390688I  52 

9 

576379 

5.17 

966705 

.86   609674 

6-o3 

890826 

389964 

5i 

.  IC 

576689 

5-16 

966653 

.86   6ioo36 

6-02 

5o 

11 

9-576999 

5-i6 

9-966602 

.861  9-610397 

6-02 

10.889608 

il 

12 

t]fa 

5-16 

966550 

•  86 

610739 

6-02 

389241 
388880 

i3 

5.i5 

966499 

.86 

611120 

6-01 

47 

14 

577927 

5-15 

966447 

.86 

611480 

6-01 

388520 

46 

i5 

578236 

5.14 

966895 

.86 

611841 

6-01 

888159 

45 

i6 

578545 

5.14 

966344 

.86 

612201 

6-00 

887799 
887489 

44 

\l 

578853 

5.i3 

966292'  .86 

6i256i 

6-00 

43 

579162 

5-i3 

966240I  .86 

612921 

6-00 

387079 

42 

19 

579470 

5-13 

966188!  .86 

613281 

5-99 

886719!  41 

20 

579777 

5-12 

966136 

.86 

618641 

til 

386359I  40 

21 

V58oo85 

5-12 

9-966085 

.87 

9-614000 

10.386000  89 
385641  38 

22 

580392 

5-11 

966033 

.87 

614359 

5.98 

23 

580699 

5.11 

960981 

.87 

614718 

5.98 

385282  37 
884928  36 
384365!  85 

24 
25 

58I00D 
58i3i2 

5. II 
5.10 

96'')928 
965876 

.87 
.87 

6i5o77 
613435 

5-97 
5.97 

26 

58i6i8 

5.10 

960824 

.87 

615798 

5.97 

884207!  34 

11 

581924 

5.09 

965772 

.87 

6i6i5i 

5-96 

888849I  33 

582229 

5.09 

963720 

.87 

6i65o9 

5.96 

388491!  32 
388i88j  3i 

^9 

582533 

t:i 

963668 

.87 

616867 

5.96 

3o 

582840 

965613 

.87 

617224 

5.95 

382776  3o 

3i 

9-583145 

5.08 

9-963363 

.87 

9-617582 

5.95 

10-882418 

11 

32 

583449 

5.07 

9655 1 1 

.87 

617989 

5.95 

382061 

33 

583754 

5.07 

965458 

.87 

618293 
6i8652 

5.94 

381705  27 

34 

584058 

5.06 

965406 

il 

5-94 

381848!  26 

35 

584361 

5.06 

965353 

619008 

5.94 

880992  25 
380686!  24 

36 

584665 

5.06 

965301 

.88 

619864 

5-93 

ll 

584968 

5.o5 

965248 

.88 

619721 

5-98 

380279!  23 

585272 

5.o5 

965195 

.88 

620076 

5.93 

879924!  22 

39 

585574 

5-04 

965143 

.88 

620482 

5.92 

379368!  21 

40 

585877 

5-o4 

905090 
9-965087 

.88 

620787 

5.92 

879218  20 

4i 

9-586179 

5.o3 

.88 

9-621142 

5.92 

10-378858 

;? 

42 

586482 

5.o3 

964984 

.88 

621497 
621852 

5.91 

878508 

43 

586783 

5-o3 

96493 1 

.88 

5.91 

878148 

17 

44 

587085 

5.02 

964879 

.88 

622207 

5.90 

3777981  16 
877489!  i5 

45 

587386 

5-02 

964826 

.88 

622561 

5.90 

46 

587688 

5.01 

964773 

.88 

622915 

IX 

877085;  14 

s 

587989 

5.01 

964719 

.88 

623269 
628628 

376781!  i3 

588289 

5-01 

964666 

.89 

s'o^ 

876877;  12 

jg 

588590 

5.00 

964613 

.89 

628076 

376024,  11 

1)0 

588890 

5.00 

964560 

.89 

624880 

875670,  10 

5i 

9-589190 
589489 

4-99 

9-964507 

.89!  9-624683 

5.88 

10.375317!  9 
374964!  8 

52 

4.99 

964454 

.89!   625o36 

5.88 

53 

'^l^ 

4-99 
4.98 

964400 

.89I   625388 

5.87 

3746121  7 

54 

964347 

.89!   625741 

5.87 

374259!  6 

55 

590387 

4-98 

964294 

.891   626093 
.89   626445 

5.87 

3789071  5 

56 

590686 

4-97 

964240 

5.86 

373355!  4 

u 

590984 

4-97 

964187 

.89!   626797 
.89'   627149 

5.86 

378203'  3 

591282 

4-97 
4-96 

964133 

5.86 

37285i[  2 

59 

59080 

964080 

.891   627501 

5.85 

372499   I 
372148:  0 

^ 

59187b- 

4-96 

964026 

.89!   627852 

5.85 

Cosuie 

D.      1 

Si  110 

G7°l  Cotiui<r.  1 

D. 

Tau^.  1  M. 

BINES    AND    TANGENTS.       (23    DKQHEKS.) 


41 


0 

Sine 

-^• 

Cosine   D.  1  Tang. 

1   ^• 

1  Cotan^*. 

1 

9.591878 

496 

9.964026  .891  9.627832 

!  5-85 

10.372148!  60 

1 

592176 

4-95 

963972  .89'   628203 

1  5-85 

371797!  5o 
371446I  5ft 

a 

592473 

4-95 

963919  .89;   628334 

5-85 

3 

592770 

4-95 

963863  .901   628903 

5-84 

371093!  57 

4 
5 

^^l 

4-94 
4.94 

963811  .90   629255 
963757  .90   629606 

5-84 
5-83 

370743 
370394 

56 
55 

6 

593659 

4.93 

963704  .90   629936 

5-83 

370044 

54 

I 

593953 

4.93 

963630  .90!   63o3o6 

5-83 

369694 

53 

5^251 

493 

963596  .90 

63o656 

5-83 

369344 

52 

9 

594547 

4-92 

963342!  .90 

63ioo5 

5-82 

36«995 

5i 

10 

594842 

4-92 

963488;  .9c 

631355 

5-82 

368643 

5o 

11 

9.595137 

4-91 

9-963434!  -90 

9-63i7o4 

5-82 

10-368296 

49 

12 

^  595432 

4-91 

963379  .90 

632033 

5-8. 

367947 

48 

i3 

595727 

491 

963323,  .90 

632401 

5-81 

367399 

47 

U 

596021 

4.90 

963271 1  .90 

632750 

5-81 

367230 

46 

i5 

5963 1 5 

iX 

9632171  .90 
963i63|  .90 

633098 

5-80 

366902 1  43  1 

i6 

^ 

633447 

5-80 

366553 

44 

\l 

4.89 

963io8j  -91 

633793 

5-80 

366203 

43 

597196 

4.89 

963034'  -91 

634143 

5-79 

365837 

42 

19 

n\x 

4-88 

962999  .91 

634490 

5.79 

365310 

41 

20 

4-88 

962943:  .91 

634838 

i:?I 

363162 

40 

21 

9.598075 

4-87 

9-962800!  -gt 
962836'  .91 

9-633183 

10-O64815 

^ 

22 

^  598368 

4.87 

635532 

5-78 

304468 

23 

598660 

4.87 

962781 

.91 

635879 

5.78 

364I2I 

^2 

24 

598952 

4.86 

962727 

.91 

636226 

5-77 

363774 

23 

599244 

4-86 

962672 

•91 

636572 

5-77 

363428 

35 

26 

599536 

4-85 

962617 

•91 

636919 

5-77 

363o8i 

34 

?2 

599827 

4-85 

962362 

.91 

637263 

5-77 

362735 

33 

600118 

4-85 

962308 

•91 

63761 1 

5-76 

362389 

32 

29 

600409 

4.84 

962433 

•91 

'dlt 

5.76 

362044 

3i 

3o 

600700 

4-84 

962398 

•92 

5.76 

361698 

3o 

3i 

9.600990 

4-84 

9-962343 

•92 

9-638647 

5.75 

io-36i333 

29 

32 

^  601280 

4.83 

962288 

•92 

t^', 

5.75 

36ioo8 

28 

33 

601370 

4-83 

962233 

•92 

5.75 

36o663 

27 

34 

60 r 860 

4-82 

962178 

•92 

639682 

5-74 

36o3i8 

26 

35 

602 1 5o 

4.82 

962123 

•92 

640027 

5-74 

339973 

25 

36 

602439 

4.82 

962067 

•92 

640371 

5.74 

339629 

24 

ll 

602728 

4-81 

962012 

•92 

640716 

5.73 

359284 

23 

6o3oi7 

4.81 

961957 

•92 

641060 

5.73 

338940 

22 

39 

6o33o5 

4-81 

961002 
961846 

•92 

641404 

5.73 

358396 

21 

40 

603594 

4-80 

•92 

641747 

5-72 

358233 

20 

4f   9-60.3882 

4-8o 

9-961791 

•92 

9-642001 
642434 

5-72 

lo- 357909 
357566 

\l 

4:    604170 

4-79 

961735  -92 

5-72. 

43 

604457 
604745 

4-79 

961680;  -92 

642777 

5-72 

337223,  17 
356880  16 

44 

t]l 

961624!  -93 

643120 

5-71 

45 

6o5o32 

961369'  -93 
96i5i3  -93 

643463 

5-71 

3565371  i5 

46 

6o53i9 
6o56o6 

4-78 

643806 

5-71 

356I04I  14 

s 

4-78 

961458;  -93 

644148 

5-70 

335832J  i3 

605892 

4-77 

961402'  -93 

644490 
644832 

5-70 

3555io!  12 

49 

606179 
606465 

t]l 

961346^  -93 

5-70 

355i68  11 

5o 

961290!  -93 

645174 

5.69 

354826'  10 

5i 

9.606751 

4-76 

9-9612351  -93  9.645516 

5-69 

10.3544841  9 

52 

607036 

4-76 

961179!  -93!   645357 
961123!  -93;   646199 

5.69 

354143!  8 

53 

607322 

4-75 

t^ 

353801   7 
3534601  6 

54 

607607 

4-75 

961067  -93'   646340 

55 

607892 

4-74 

961011  -93!   646881 
960955  -93!   647222 
96o890'  .93!   647562 
960843,  -94   647903 

5-68 

3531IQI  5 

332775!  i 

56 

608177 

4-74 

5-68 

U 

608461 

4-74 

5-67 

352438  3 

608745 

4-73 

5-67 

352097 

2 

59 

609c 20 
6o93i3 

4.73 

960786  -941   648243 

5.67 

351737 

I 

60 

4.73 

9607301  -9^   648583 

5.66 

351417 

0 

Cosino 

D. 

Sine  Ica-I  Cotang. 

D. 

hmg.    [ 

-iL. 

42 

(24 

DEGREES.)   A 

TABLE  01?  LOOARITHMTO 

M. 

0 

Sine 
9.609313 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

60 

4 

73 

9-960730 

•94 

9-648583 

5.66 

io-35i4i7 

I 

609597 

4 

72 

960674 

.94 

648923 

5-66 

351077 

^ 

3 

609880 

4 

72 

960618 

.94 

649263 

5-66 

350737 

3 

610164 

4 

72 

960561 

-94 

649602 

5-66 

350398 

37 

4 

610447 

4 

71 

96o5o5 

•94 

649942 

5-65 

350038 

56 

5 

610729 

4 

71 

960448 

•94 

65o28i 

5-65 

3497 '9 

55 

6 

611012 

4 

70 

960392 
960335 

.94 

65o62o 

5-65 

349380 

54 

I 

61 1294 

4 

70 

.94 

650959 

5-64 

349041 

348703 

53 

611576 

4 

70 

960279 

.94 

65 1 297 

5-64 

52 

9 

6ii858 

4 

69 

960222 

.94 

65 1 636 

5-64 

348364 

5i 

IC 

61 2140 

4 

69 

960165 

.94 

651974 

5-63 

348026 

5o 

II 

9-612421 

4 

69 

9-960109 

-95 

9-652312 

5-63 

10-347688 

8 

12 

612702 

4 

68 

960052 

.95 

65265o 

5-63 

347350 

i3 

612983 

4 

68 

959995 
959938 

-95 

652988 

5-63 

347012 

47 

I^ 

6i3264 

4 

67 

.95 

653326 

5-62 

346674 

46 

I') 

613545 

4 

67 

959882 

-95 

653663 

5-62 

346337 

45 

I6 

6i3825 

4 

67 

959825 

.95 

654000 

5-62 

346000 

44 

;^ 

6i4io5 

4 

66 

959768 

•95 

654337 

5-61 

345663 

43 

614385 

4 

66 

939711 

-95 

654674 

5-6i 

345326 

42 

19 

614665 

4 

66 

959654 

-95 

6550II 

5-61 

3449S9 

41 

20 

614944 

4 

65 

959596 

•95 

655348 

5.61 

344652 

40 

21 

9-6i5223 

4 

65 

9-939539 

-95 

9-655684 

5.60 

ID -3443 1 6 

39 
38 

2i 

6i?0o2 

4 

65 

939482 

-95 

656020 

5.60 

343980 

23 

6I5^8I 

4 

64 

959425 

.95 

656356 

5.60 

3-43644 

37 

24 

616060 

4 

64 

959368 

•95 

636692 

5.59 

343308 

36 

25 

6 1 63 38 

4 

64 

959310 

.96 

657028 

5.59 

342972 

35 

26 

616616 

4 

63 

959253 

657364 

5-59 

342636 

34 

52 

616894 

4 

63 

959195 
959138 

.96 

658o?4 

342301 

33 

617172 

4 

62 

.96 

341966 

3«s 

?9 

617430 

4 

62 

959081 

.^6 

658369 

5^58 

34i63i 

3i 

3o 

617727 

4 

62 

939023 

.96 

658704 

5-58 

341296 

3o 

3i 

9-618004 

4 

6i 

9-958965 

.96 

't^',^ 

5-58 

10-340961 

20 
28 

32 

618281 

4 

61 

938908 

.96 

5.57 

340627 

33 

6i8558 

4 

6i 

958850 

.96 

659708 

5.57 

340292 

27 

34 

618834 

4 

60 

958792 

.96 

660042 

5.57 

339938 

26 

35 

61QI10 

4 

60 

958734 

.96 

660376 

5.57 

339624 

25 

36 

619386 

4 

60 

958677 

.96 

660710 

5-56 

339290 

24 

3i 

619662 

4 

59 

958619 

-96 

661043 

5-56 

338937 

23 

3^ 

619938 

4 

59 

958561 

-^6 

66.377 

^/^i 

338623 

22 

3g 

620213 

4 

u 

9585o3 

-97 

661710 

5-55 

338290 

21 

40 

620488 

4 

958445 

-97I   662043 

5-55 

337937 

20 

41 

9-620763 

4 

58 

9-958387 

-97  9-662376 

5.55 

10-337624 

19 

4a 

621038 

4 

57 

938329 

•97 

662709 

5.54 

Vj^^^i 

43 

62i3i3 

4 

57 

958271 

•97 

663042 

5.54 

336938 

n 

44 

621587 

4 

ll 

958213 

•97 

663375 

5.54 

336625 

16 

45 

621861 

4 

958 1 54 

-97 

663707 

5.54 

336293 

i5 

46 

622135 

4 

56 

958096 
938o38 

•97 

664039 

5-53 

335961 

14 

ii 

622409 

4 

56 

•97 

664371 

5.53 

335629 

i3 

622682 

4 

55 

957979 

•97 

664703 

5-53 

335297 

12 

49 

622956 

4- 

55 

957921 

•97 

665o35 

5.53 

334965 

II 

56 

623229 

4 

55 

957S63 

•97 

665366 

5-52 

334634 

10 

5i 

9-6235o2 

4 

54 

9-957804 

•97 

9-665697 

5.52 

10 -334303 

t 

52 

623774 

4 

54 

937746 

-98 

666029 

5-52 

333971 

53 

-624047 

4 

54 

957687 

-98 

666360 

5-51 

333640 

I 

54 

624319 

4 

53 

957628 

.98 

666691 

5.5i 

333309 

55 

624591 

4 

53 

957570 

-98 

667021 

5.5i 

332979 

5 

56 

624863 

4 

53 

95751 1 

.98 

667352 

5-51 

332648 

4 

U 

625i35 

4 

52 

957452 

.98 

667682 

5-5o 

3323i8 

3 

625406 

4 

52 

957393 

.98   668013 

5.50 

331987 

2 

59 

625677 

4 

52 

937335 

.98   668343 

5-5o 

33i65t 

1 

66 

625948 

4 

5i 

937276 

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5-5o 

33i328 

0 

L  _ 

Cosine 

1 

»• 

Sine  1 

GS^i  Cotiuif?. 

T). 

TfUT£:._ 

A-- 

SINES    AND    TANGENTS.       (26    DEOREiSS.) 


43 


M. 

Bino 

D. 

Cosine  | 

D.| 

Tang. 

^•_ 

Ct)tang. 
10.331327 
33oQ9b 

o 

9-625948 

4-5i 

9.957276 

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9.668673 

5 

5o 

60 

626219 

4-5i 

957217; 

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669002 

5 

49 

5I 

a 

626490 

4-31 

5571 58 

.98 

669332 

5 

49 

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3 

626760 

4-50 

937099 

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669661 

5 

49 

48 
48 

330339 

ii 

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627300 

4-5o 
4-5o 

937040 
936981 

t 

670320 

5 

5 

330009 
329680 

56 
55 

6 

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4.49 

93692 1 

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5 

48 

32935,  54 

I 

627840 
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4-49 
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5 
5 

48 
47 

329023  53 
328694  52 

9 

62837b 

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5 

47 

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5i 

10 

628647 

4-48 

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5 

47 

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II 

9'6289i6 

4-47 

Q.936625I 

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9-672291 

5 

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it 

12 

629185 

4-47 

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9365o6 

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46 

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14 

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4-46 

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3 

46 

326726 

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46 

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674584 

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45 

325416 

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63 1 039 

4-45 

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674910 

5 

44 

325090 

41 

20 

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4-45 

936089  I 

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673237 

5 

44 

324763 

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21 

9-63i593 

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9.936029  I 

•  00 

0.675504 

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44 

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It 

22 

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4.44 

93.V.9  I 

•  00 

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44 

324110 

23 

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•  00 

676216 

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43 

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24 

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4-43 

953849  J 

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6-^6543 
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43 

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25 

4-43 

955789  1 

-00 

5 

43 

323i3i 

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26 

632923 

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953729  1 

•  00 

677 '94 

5 

43 

322806 

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11 

633189 

4-42 

953669  I 

•  00 

677520 

5 

42 

322480 

33 

633454 

4-42 

9336091 

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677846 

5 

42 

322154 

32 

29 

633719 

4-42 

955548  I 

-00 

678171 

5 

42 

321829 

3i 

3o 

633984 

4-41 

955488  I 

•  00 

678496 

5 

42 

32i5o4 

3o 

3i 

9-634249 

4-41 

9.935428  1 

•01 

9.678821 

5 

41 

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320834 

29 

32 

6345.4 

4-40 

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41 

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33 

634778 

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41 

320529 

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34 

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320203 

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6353o6 

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933186  I 

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40 

319880 

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955126  I 

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40 

319536 

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3.8908 

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681416 

5 

39 

3.8384 

21 

40 

636623 

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318260 

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42 

637148 

4.37 

954762  I 

-01 

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5 

l^ 

3176.3 

43 

637411 

4-37 

954701  I 
954640  I 

•01 

6S2710 

5 

317290 

17 

44 

637673 

ill 

•  01 

6S3o33 

5 

38 

3.6967 

16 

45 

637935 

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954518  I 

-c: 

6S3356 

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38 

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14 

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4-36 

954457  > 

•  02 

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5 

37 

315999 

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638730 

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954396  I 

-02 

6^4324 

5 

37 

3.5676 

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1  49 

638981 

4-35 

9543351 

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684646 

5 

37 

3.5354 

>  56 

639242 

4-35 

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•  02 

6H4968 

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3i5o37 

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9 -639503 

4-34 

9-954213  I 

•  02 

g  685290 

5 

r/%.3147101  Q 
J 1 4388  8 

52 

639764 

4.34 

954152  I 

•02  '  6S5612 

5 

36 

53 

640024 

4-34 

954090  I 

•02!   6S5934 

5 

36 

3.4066  7 

54 

640284 

4-33 

954029  I 
953968  I 

-021   6S6255 

5 

36 

3i3745i  6 

55 

640344 

4-33 

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•02   6H6893 

5 

35 

3i3423!  5 

56 

640804 

4-33 

953906  I 
953845  I 

5 

35 

3i3io2:  4 

U 

641064 

4-32 

•02   681219 

5 

35 

312781J  3 

641324 

432 

953783  I 

•02I   687340 

5 

35 

3i2l6o  2 

59 

641584 

4-32 

953722  I 

•o3    (,HiHb\ 

5 

-34 

312.39  1 

~- 

641842 

4-3i 

9536601 

•o3 

688182 
,  Cotan^. 

5 

.34 

3ii8i8 

1. 

Cosine 

D. 

Sine   t 

140 

□ 

D.'  ~ 

1  Tang. 

a 

(26 

DEGUEES.)   A 

I'ABLB  OF  LOUARITnMIC 

'^r 

Siiio 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9 •641842 

4-31 

9 ■ 953660 

r^3 

9.688182 

5.34 

io.3ii8i8 

60 

1 

642101 

4.31 

im 

o3 

688502 

5 

34 

311498 

5q 

2 

642360 

4-31 

o3 

688823 

5 

34 

311177  58 

3 

642618 

4-3o 

953475 
953413 

o3 

689143 

5 

33 

3 108571  57 

i 

642877 

4-3o 

o3 

689463 

5 

33 

310537;  56 

5 

6431 35 

4 -30 

953352 

o3 

689783 

5 

33 

310217;  55 

6 

643393 

4-3o 

953290  I 

o3 

690103 

5 

33 

309897;  54 

I 

643630 

4-29 

o53228 

o3 

690423 

5 

33 

30^258  52 

643908 

4-29 

q33i66 

o3 

690742 

5 

32 

Q 

644165 

4-29 

q53io4 

o3 

691062 

5 

32 

308938  5 1 

lo 

644423 

4-28 

953042 

o3 

691381 

5 

32 

308619  5o 

II 

9.644680 

4  28 

g. 952980 

04 

9.691700 

5 

3i 

io.3o83oo;  49 
307981  48 
307662  47 

12 

6X4936 

4  28 

952918 

04 

692019 

5 

3i 

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645193 

4  27 

952855 

04 

692338 
692656 
692975 

5 

3i 

14 

6454D0 

4  27 

952793 
932731 

04 

5 

3i 

807344  46 

i5 

645706 

4  27 

I 

04 

5 

3i 

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45  1 

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4  26 

932669' I 

04 

693293 

5 

3o 

306707 
3o6388 

44 

11 

646218 

4  26 

952606  I 

04 

693612 

5 

3o 

43 

646474 

4 

26 

952544  I 

04 

693930 

5 

3o 

306070 
3o5752 

42 

•9 

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4 

25 

952481 

04 

694248 

5 

3o 

41 

20 

646984 

4 

25 

952419 

04 

694566 

5 

29 

3o543ii  40 

21 

9-647240 

4 

25 

9.952356 

04 

9-694883 

i 

29 

io.3o5n7!  3q 
304799!  38 
304482 1  37 

22 

647494 

4 

24 

952294 
952231 

04 

695201 

5 

29 

23 

647749 

4 

24 

04 

693318 

5 

29 

24 

648004 

4 

24 

952168 

o5 

695836 

5 

11 

304164!  36 

25 

648208 

4 

24 

952106 

o5 

696153 

5 

3o3847i  35 

26 

648012 

4 

23 

952043 

o5 

696470 

5 

28 

3o353o^  34 

27 

648766 

4 

23 

951980 

o5 

696787 

5 

28 

3o32i3:  33 

2d 

649020 

4 

23 

931917 

o5 

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5 

28 

3028971  52 

29 

649274 

4 

22 

951 854 

o5 

697420 

5 

27 

3j258oI  3i 

3o 

649327 

4 

22 

951791 

1 

o5 

697736 

5 

27 

302264  3o 

3i 

9-649781 

4 

22 

Q. 93 1728 

o5 

9.698033 

5 

27 

10.3019471  29 
3oi63ii  28 

32 

65oo34 

4 

22 

^   931665 

o5 

698369 

5 

27 

33 

650287 

4 

21 

901602 

1 

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5 

26 

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300999!  26 
3oo6G4i  25 

34 

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4 

21 

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69900 1 

5 

26 

35 

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4 

21 

951476 

o5 

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5 

26 

36 

65 1 044 

4 

20 

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699632 

5 

26 

3oo368;  24 

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4 

20 

951 349 

06 

699947 

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26 

3ooo53i  23 

38 
39 

65 1 549 
65 1 800 

4 
4 

20 
19 

951286 
951222 

06 
06 

700263 
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5 
5 

25 
25 

2997371  23 
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40 

652052 

4 

19 

95,159 

06 

70c 893 

5 

25 

299107 

20 

41 

«.6523o4 

4 

\t 

Q. 951096 

1 

06 

g.  701208 

5 

24 

10.298792 

\l 

42 

652555 

4 

95io32 

I 

06 

701523 

5 

24 

298477 

43 

652806 

4 

18 

950968 

06 

701837 

5 

24 

298163 

n 

44 

653o57 

4 

18 

950905 

I 

06 

702152 

5 

24 

297848 

16 

45 

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4 

18 

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1 

06 

702466 

5 

24 

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4 

17 

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06 

702780 

5 

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17 

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06 

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5 

23 

296^05!  1 3 

48 

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5 

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49 

634309 

4 

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06 

5 

23 

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5o 

654558 

4 

16 

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07 

704036 

5 

22 

295g64i  10 

5i 

9.654808 

4 

16 

9.9504581 1 

07 

9.704350 

5 

22 

10.295650;   9 

295337I  8 
295o23j  7 

52 

655o58 

4 

.16 

950394  I 
95o33o  I 

07 

704663 

5 

22 

53 

655307 

4 

i5 

07 

704977 

5 

22 

54 

655556 

4 

i5 

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07 

705290 

5 

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55 

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4 

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07 

703603 

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21 

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294084  4 

56 

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4 

14 

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07 

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5 

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6563o2 

4 

14 

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07 

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5 

21 

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07 

706541 
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5 

21 

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59 

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1  4 

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07 

5 

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298146;  I 
2928341  0 

66 

1   65704 ; 
Cosin'j 

1  4 

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949881  1-07 

707166 

5-20 

1  ~; 

D. 

Sijje  !63=^ 

Cotang. 

D. 

Taiig. 

M-J 

BINES  ANli  TANGENTS 

(27  DEGREES. 

) 

60 

MT 

Sino 

D. 

Cosine  1  D. 

1  Tang. 

D. 

Cotang. 

0 

•>fp 

4-i3 

9-949881  1-07 

j  9-707166 

5-20 

10-292834 

I 

4-i3 

949816  1-07 

1   707478 

5-20 

2925i2|  5o 

a 

657542 

4-12 

949732  1-07 

707790 
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5-20 

292210  5b 

3 

6577O0 

'   4-12 

949688  1-08 

5-20 

t^,  u 

i. 

658o37 

4-12 

9I9623  1-0^3 

1   70S414 

5-19 

t 

658284 

!   4-12 

949558  I -08 

1   708726 

5-19 

291274  55 

6 

658531 

1  4-II 

949494  1- 08 

709037 

5-19 

2909631  54 

I 

658778 

4-II 

949429  \-oi 

709349 

5-19 

29065 1  53 

659025 

4-11 

949364  I -08 

709660 

t\l 

290340  5a 

9 

659271 

4-10 

949300  1-08 

1   709971 

2?97i8'  5a 

10 

659517 

'  4-10 

949235|i-oS 

1   710282 

5-18 

11 

9-659763 

4-10 

9-949170  I-O^ 

j  9-710593 

5.18 

288785 

% 

47 
46 

la 
i3 

660009 
660255 

4-09 
4-09 

949105  I- oi 
949040  I  -oS 

7 '0904 
1   711215 

5.18 
5.18 

14 

66o5oi 

4-09 

948975  I -08 
948010  I -08 

i   711525 

5.17 

288475 

i5 

660746 

4-09 

711836 

5-17 

288164 

45 

i6 

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4-o8 

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712146 

5-17 

287854 

44 

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661236 

4-oS 

948780;  I -09 

712456 

5-17 

287544 

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4-o8 

948715  1-09 

712766 

5.16 

287234 

42 

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661726 

4-07 

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713076 

5-16 

286924 

41 

30 

661970 

4-07 

948584  1-09 

713386 

5-16 

286614 

40 

ai 

9-662214 

4-07 

9-948519,1  -09 

9.713696 

5-16 

10-286J04 

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27 

662459 
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4-07 

9484541-09 

714005 

5.16 

285995 

23 

4-o6 

948388,1-09 

714314 

5.i5 

285686 

u 

24 

662946 

4-06 

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714624 

5.i5 

285376 

25 

663190 
6634i3 

4-o6 

948257 

1-09 

714933 

5.i5 

285067 

35 

26 

4-o5 

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1-09 

715242 

5-15 

284758 

34 

J2 

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4-o5 

948126 

1-09 

7i555i 

5.14 

284449 

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5-14 

284140 

32 

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4-o5 

947995  >-'o 

716168 

5.14 

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4-04 

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7 '6477 

5-14 

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9-664648 

4-04 

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l-IO 

9-716785 

5-14 

10.283215 

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32 

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4-04 

947797 
947731 

1  -10 

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5-13 

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33 

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717401 

5-13 

282D99 

27 

34 

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26 

35 

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4 -03 

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5.i3 

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36 

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947533  I -10 

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281670 

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4-02 

947467!  I -10 

718633 

5-12 

281367 

23 

666342 

4-02 

947401 |i- 10 

718940 

5-12 

281060 

22 

39 

666583 

4-02 

9473351-10 

719248 

5-12 

280752 

21 

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666824 

4-01 

9472601-10 
9-947203,1-10 

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5-12 

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20 

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9-667065 

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9.719862 

5-12 

io.28oi38 

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42 

667305 

4-01 

947i36ji-ii 

720169 
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5-11 

279831 

43 

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4-01 

947070  I -II 

5-11 

279524 

17 

44 

667786 

4-00 

947004  I -11 

720783 

5-II 

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16 

45 

668027 

4-00 

946037  III 
946871  I -11 

721089 

5-11 

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46 

668267 

4-00 

721396 

5-11 

278604 

14 

% 

6685o6 

3.99 

946804  I'll 

721702 

5-10 

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668746 

3-99 

946738  I  - 1 1 

722009 

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5-10 

277901 

12 

49 

66S986 

3-99 

946671  I -11 

5-10 

277685;  II 

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722621 

5-10 

277379  10 

10. 277073  9 

276768  8 

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9.660464 

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9-722927 

5-10 

52 

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5.09 

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3.97 

946337  I- M 

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5.09 

55 

670419 
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275851   5 

56 

3.97 

9462031- 12 

724454 

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275546  4 

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670806 

3-97 

946i36!i-i2 

724759 
725o6d 

275241   3 

671 i34 

3.96 

946069  1-12 

5.08 

274935  2 

J? 

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3-96 

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725369 

5  08 

274631   I 

6o 

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3-96 

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725674 

5.08 

274326  0 

Coeiiie 

D. 

Sme  i62o 

Cbtaiw?. 

D   1 

Taiig.  [ 

Tl 

46 


(28    DKORKES.)       A    TABLE    OP    LOG  A-RnfiMIfJ 


M. 


9 

10 

II 

13 

i3 
14 
i5 
i6 

\l 

19 

20 
31 
32 
23 
24 
35 
36 

aS 

29 

3o 
3i 

33 

33 
34 
35 

36 

3^^ 

39 
40 
41 

42 

43 
44 
45 
46 

47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

u 

5q 
6o 


Sine  I 


D. 


9*671609 

671847 

672084 

672321 

672553 

672795 

673032 

673268 

673505 

67374 

67397 

9-674213 
6744/18 
674684 
674919 
675155 
675390 
675624 
675859 
676094 
676328 

9-676562 
676796 
6770J0 
677264 
677498 
677731 
677964 
678197 
678430 
678663 

9-678895 
679128 
679360 
679592 
679824 
68oo56 
680288 
68o5i9 
680750 
680982 

9-68i2i3 
681443! 
681674 
681905 
682135 
682365 
682595 
682825 
683o55 
683284 

9-683514 
683743 
683972 
684201 
684430 
684658 
684887 
685ii5i 
6853431 
685571 


3.96 
3.95 
3.95 
3.95 
3.95 
3.94 
3.94 
3.94 


94 
93 

93 

93 

92 

92 

92 

92 

91 

91 

91 

91 

3' go 

3  90 

3.90 

3.Q0 


CoBino  1  D. 


Coftine  I   D. 


9-945o35  1.12 
945868  1. 1 2 
945800  I  •  1 2 

945733  1-12 

945666  I 
945598  I 
945531  I 
945464  I 
945396  I 
945328  I 
945261  I 

9-945193  I 
945125  I 
945o58  I 
944990  I 
944922  I 
944854  I 
944786  I 
944718  I 
944650  I 
944582  I 

9-944514  I 
944446  I 
944377  I 
944309  I 
944241,1 
9441721 
944104  I  . 
944o36  1-14 
94396711-14 
943899!!  .14 

9-943830  I  - 14 
943761 1 1- 14 
943693  I  - 1 5 
943624  I- 15 
943555,1-15 
943486  1-15 
943417  I- 15 
943348  i-i5 
943279  I -15 
943210  1-15 

9-943141  i-i5 
943072  I -15 
943oo3  I -15 
942934  I- 1 5 
942864  I- 1 5 
942795  I -16 
942726  I- 16 
942656  i-i6 
942587  1-16 
942517  i.i6 

9-942448  I 
942378  I 
942308;  I 
942239  I 
942169  I 
942099  I 
942029  I 

94i9'^9  I 
941889  1-17 
941819  1-17 


Tang. 


Sine 


9-725674; 
725979' 

726284! 
726588. 
726892! 

727197: 
727501 
727805 
728109' 
7284121 
728716I 

9.729020I 
729323; 
729626I 
729929' 
73o233i 
73o535: 
73o838 
731141 
731444' 
731746' 

9-732048, 
73235il 
732653; 
732955J 
733257I 
733558| 
7338601 
734162! 
7344631 
734764! 

9- 735066 I 
735367 
735668! 
7 3 5969 I 
736269! 
736570 
736871 
737171 
737471 
737771 

9-738071 
738371 
738671 
738971 
739271 
739570 
739870 
740169 
740468 
740761 

9.741066 
741365 
741664 
741962 
742:61 
742559 
742858; 
743i56j 
743454! 
743753] 

Cctang.  _[ 


D. 

Cotang. 

1 

5-08 

10-274336 

'6^ 

5.08 

274021 

It 

5.07 

273716 

5.07 

273412 

u 

5-07 

273108 

5.07 

272803 

55 

5-07 
5.06 

27249Q 
272195 

54 

53 

5-06 

271891 

52 

5.06 

271588 

5i 

5-06 

271284 

5o 

5-06 

10-270980 

1? 

5-o5 

270677 

5-o5 

270374 

47 

5-o5 

270071 

46 

5-o5 

269767 

45 

5-05 

269465 

44 

5-04 

269162 

43 

5-04 

268859 

41 

5.04 

268556 

41 

5-04 

268254 

40 

5-04 

10-267952 

It 

5-o3 

267649 

5-o3 

267347 

37 

5-o3 

267045 

56 

5-o3 

266743 

35 

5-o3 

266442 

34 

5-02 

266 1 40 

33 

5-02 

265M38 

32 

5-02 

265537 

3i 

5-02 

265236 

3o 

5.02 

10-264934 

It 

5-02 

264633 

5-01 

264332 

27 

5-01 

26403 1 

26 

5-01 

263731 

25 

5-01 

26343o 

24 

5-01 

263129 

23 

5-00 

262829 

22 

5-00 

262529 

21 

5 -00 

262229 

20 

5-00 

10-261929 

\l 

5.00 

261629 

4-99 

261329 

17 

4-99 

261029 

16 

4-99 

260729 

i5 

4.99 

260430! 

14 

4.99 

260 i3o 

i3 

4.90 
4-98 

2598311 

12 

259532 

11 

4-98  1 

259233, 
10.258934! 

10 

4-98 

? 

4-98 

258635! 

4-98 

2583 36i 

I 

4-97 

268o38: 

4-97 

257739: 

4-97 

257441 

4-97 

257142 

4-97 

256844 

4-97 

356546; 

4-96 

256248. 

0 

D.   1 

__Tan^  J^ 

J^!L 

8INE8  AND  TAN0KNT8. 

(29  DKOIIBES.^ 

> 

47 

M.' 

0 

Sine 
9-685571 

D. 

Cosine  1  D.  1  Tang. 

D. 

CoUing.  I 

3.80 

9-9418191.171  9-743752 

496 

10-256248,  60 

I 

685799 

3.79 

9417491-17!   744o5o 

4.96 

355950!  59 
2556521  58 

3 

686027 

3-79 

94.679  .•.71   744348 

4-96 

3 

686254 

3.79 

94.6091-171   744645 

4-96 

2553551  57 

4 

686482 

i^ 

94.539  I -17I   744943 

4.96 

255o57  56 

5 

^5? 

94.4601-171   743240 
94.3981-17!   745538 

4.96 

254760J  55 

6 

3-78 

4-95 

254462  54 

I 

687163 

3.78 

94.3281-171   745835 

4-95 

254.65 

53 

687389 

3.78 

94.2581-171   746132 

4-95 

253868 

52 

9 

687616 

3-77 

941187  '-n:  746429 

4-95 

253571 

5i 

10 

687843 

3-77 

941 1 17 1 -ni  746726 

4-95 

253274 

5o 

II 

"■^ 

3-77 

9.941046  1-18  9-747023 

4.94 

10-252977 
25268. 

il 

la 

3.77 

9409751-18:  7473.9 

4-94 

id 

688521 

3-76 

940005.1-18  747616 

4.94 

2523S4 

^J 

14 

688747 

3-76 

940834  I -181   7479>3 

4.94 

252087 

46 

15 

688972 

3-76 

Q40763  1-18 

748209 
7485o5 

4-94 

25.79. 

45 

i6 

689198 

3.76 

940693  1-18 

4-93 

23.495 

44 

\l 

689423 

3-75 

940622,1-18!   748801 

493 

25o9o3 

43 

689648 

3.75 

94055 1 1 1-18   749097 
94o48oji.i8|   749303 
9404001-181   749689 
9-94o338ji-i8,  9-749983 
940267;  I -18   750281 

4-93 

42 

19 

689873 

3.75 

4-93 

2 50607 

4. 

20 
21 

690098 
9  690323 

3.75 
3-74 

4-93 
4-93 

25o3.. 
io-i5oo.5 

40 
^2 

22 

690548 

3.74 

4-92 

249719 

38 

23 

690772 

3.74 

9401961-18   750576 

4-92 

249424 

37 

24 

690996 

3  74 

940.25:1-19   750872 

4-92 

249.28 
248833 

36 

25 

691220 

3-73 

94oo34{i-i9i   751.67 

4-92 

35 

26 

691444 

3.73 

939982  1.19 

751462 

4-92 

248538 

34 

27 

691668 

3-73 

939911  I -19 

751757 

4-92 

248243  33 

28 

691892 

3-73 

939840  1-19 

752032 

4-91 

247948  32 

29 

692115 

3-72 

939768,1-19 

752347 

4-9' 

247653!  3i 

3o 

692339 

3.72 

939697,1-19 

752642 

4.91 

247358,  3o 

3. 

9-692562 

3-72 

9.939625,1-19 

9.752937 

4-9' 

10-2470631  2n 

246769  :8 

32 

692785 

3-7. 

939354  1  - 19 

753231 

4-9' 

33 

693008 

3-71 

939482  1-19 

753526 

4-9» 

2464-4!  27 

34 

693231 

3-71 

9394.0  1-19 

753820 

4-90 

246180 

20 

35 

693453 

3.7. 

939339  1  - 19 

7541.5 

4.90 

245885 

25 

36 

693676 

3-70 

939267  1-20 
939195  1  -20 

ti^ 

4.90 

24559. 

24 

37 

693898 

3-70 

4.90 

243297 

23 

38 

694120 

3-70 

939123  l-2o[   754997 

4.90 

245oo3 

22 

39 
4o 

694342 
694564 

3-70 
3.69 

93903211 -20!    733291 

9389S0J1-20J   755585 

tx 

244709 
2444.5 

21 

20 

41 

9-694786 

3.69 

9-938908  1-20,  9.753S78 

4-89 

10-244.22 

;§ 

42 

695007 

3-69 

938836 

1-20 

756.72 

4.89 

243828 

43 

690229 

3.69 
3-68 

938763 

1-20 

756465 

4.89 

243535 

17 

44 

695450 

938691 

1  -20 

756759 

4.89 

24324. 

16 

45 

69567. 

3-68 

9386i9Ji-2o 

757052 

4-88 

242948 

i5 

46 

695892 

3-68 

9385471 « -20 
938475|i-2o 

757345 

242655 

14 

il 

696113 

3-68 

757638 

4-88 

242362 

i3 

696334 

3.67 

938402 

75793. 

4-88 

24:069 

12 

49 

696554 

3.67 

938330 

758224 

4-88 

24.776 

1. 

5o 

696775 

3-67 

938258 

7585.7 

4-88 

241483 

10 

5i 

9-696993 

3-67 

9-938i85 

9-758S10 

4-88 

10-241190 

« 

52 

6972.5 

3-66 

938  n  3 

759102 

4-87 

240898 

8 

53 

697435 

3-66 

938040 

]m; 

4-87 

24o6o5 

I 

54 

697654 

3-66 

ttl 

4-87 

24o3i3 

55 

697874 
698094 

3-66 

759979 

4-87 

240021 

5 

56 

3-65 

937822 

760272 

4-87 

239728 

4 

tl 

6983.3 

3-65 

937749 

760564 

4-87 
4-86 

239436 

3 

693532 

3-65 

937676  1-21 

760856 

23QI44 
238852 

2 

59 

69875. 

3-65 

937604  I -21 

761 148 

4-86 

I 

60 

698970 

3.64 

937531  1-21 

761439 

4-86 

238561 

0 

CoRine 

D. 

Sine   60° 

Cotang. 

D. 

Tmig. 

M. 

18 

(30 

DEGREES.)   A 

FABLE  OF  LOGARITHMIC 

ld7 

Smo 

1). 

Cosine  |  D. 

1  Tang. 

D. 

Cotang. 

\~~^ 

0 

9-698970 

3-64 

9-937531  1-21 

9-761439 

4-86 

io.23856i!  60 

I 

699189 

3-64 

937458  1-22 

761731 

4-86 

288269  50 
287077  58 

2 

699407 

3-64 

937385  1-22 

762023 

4-86 

3 

699626 

3-64 

937312  1-22 

762814 

4-86 

287685!  57 

4 

699844 

3-6: 

937238  1-22 

762606 

4-85 

287894!  56 

5 

700062 

3-oJ 

937165  1-22 

768188 

4-85 

287103  55 

6 

700280 

3-63 

937092  1-22 

4-85 

286812'  54 

I 

700498 

3-63 

937019  1  -22 

768479 

4-S5 

236521!  53 

700716 

3-63 

936046  1-22 
936872  1-22 

768770 

4-85 

286230!  52 

9 

700933 

3-62 

764061 

4-85 

235989'  5i 

10 

70i:5i 

3-62 

936799  1-22 
9-936725  1-22 

764352 

4-84 

235648 

5o 

II 

9-701368 

3-62 

9-764648 

4-84 

13-235357 

4^ 

47 

12 

i3 

701585 
701802 

3-62 
3.61 

936632  1-23 
936578  1-23 

764933 
765224 

4-84 

4-84 

235067 
284776 

14 

702019 

3-6i 

9365o5i-23 
936431 !i -23 

765514 

4-84 

284486 

46 

i5 

702235 

3.61 

7658o5 

4-84 

284195 

45 

i6 

702452 

3-6i 

936357ii-23 

766095 

4-84 

288905 

44 

]l 

702669 
702885 

3-6o 

9362841-23 

766385 

4-83 

2336i5 

43 

3-6o 

936210  1-23 

766675 

4-83 

233325 

42 

19 

7o3ioi 

3-6o 

936i36  1-23 

766965 

4-83 

233o35 

41 

20 

703317 

3-6o 

986062  1-23 

767255 

4-83 

282745 

40 

21 

9-703533 

3-59 

9-935988;! -23 

9-767545 

4-83 

10-282455 

39 

22 

703749 

3.59 

9359i4!i-23 
935840  1-23 

767884 

4-83 

282166 

38 

23 

703964 

3.59 

768124 

4-82 

281876 

37 

24 

704179 
704395 

3.59 

935766^1-24 

768418 

4-82 

23 1 587 

36 

25 

Ifs 

035692  1-24 

768708 

4-82 

281297 

35 

26 

704610 

9356i8;i-24 

768992 

4-82 

281008 

34 

11 

704825 

3-58 

935543,1-24 

769281 

4-82 

230719 

33 

7o5o4o 

3-58 

935469:1-24 

769570 

4-82 

280480 

32 

29 

705204 

3-58 

935393  1-24 

769860 

4-81 

280140 

3i 

3o 

705469 

3.57 

935320  1-24 

770148 

4-81 

220852 

3o 

3i 

9-705683 

3-57 

9-935246  I  -24 

9.770487 

4-8i 

10-229563 

11 

32 

705898 

3.57 

93517111-24 

770726 

4-8i 

229274 
228985 

33 

706112 

3-57 

935097!  1-24 

771015 

4-8i 

27 

34 

706326 

3-56 

935022 

1-24 

771808 

4-8i 

228697 

26 

35 

706539 

3-56 

934948 
934873 

1-24 

77i5o2 

4-8i 

228408 

25 

36 

706753 

3-56 

1-24 

77.8^0 

4-80 

228120 

24 

11 

706967 

3-56 

934798 

1-25 

772168 

4-80 

227882 

23 

707180 

3-55 

934723  1-25 

772457 

4-8o 

227543 

22 

39 

707393 

3-55 

934649!  1-25 

772745 

4-80 

227255 

21 

40 

707606 

3-55 

934574  1-25 

778088 

4-8o 

226967 

20 

41 

9-707819 

3-55 

9-934499  1-25 

9.778821 

4-8o 

10-226679 

'9 

18 

42 

708032 

3-54 

9344241-25 

778608 

4-79 

226892 

43 

708245 

3-54 

934349  1-25 

773806 

4-79 

226104 

17 

44 

708458 

3-54 

934274,1-25 

774184 

4-79 

2258i6 

16 

45 

708670 

3-54 

934199'! -25 

774471 

4-79 

225529 

i5 

46 

708S82 

3-53 

934123  1-25 

774759 

4-79 

225241 

14 

47 

709094 

3-53 

9340481 1-25 

775046 

4-79 

224954 

i3 

48 

709806 

3-53 

933973: 1-25 

775333 

4-79 
4-78 

224667 

12 

49 

709518 

3-53 

933898,1-26 

775621 

224879 

11 

5o 

709730 

3-53 

933822  1-26 

773908 

4-78 

224092 

10 

5i 

9-709941 

3-52 

9-933747ii-26 

9-776105 

4-78 

10  2238o5 

i 

53 

710153 

3-52 

933671  1-26 

776482 

4-78 

2 235 1 8 

53 

7io364 

3-52 

933596  1-26 

776769 

4-78 

228281 

7 

54 

710375 

3-52 

93-332o'i.26 

777o5d 

4-78 

222945 

6 

55 

710786; 

3-5i 

933445  1-26 

777342 

4  78 

222658 

5 

Go 

710997: 
71120S' 

3-5i 

933369' 1. 26 

777628 

4-77 

22J372 

4 

u 

3-5i 

933293  1 .26 

777915 

4-77 

222085 

3 

711419 

3-5i 

9332 1 7>- 26 

778201! 

4-77 

221799   2 

59 

71 1629 

3-5o 

9331 41 ;i- 26 

778487! 

4-77 

22l5l2j   I 

6o 

71 1839 

3-30 

933066; I -26!   778774I 

4-77 

221226!   0 

Cosine  1 

Ih^     1 

Si]ie_  J590]_Cotang.  [ 

D. 

_  Tang.  J 

_M.J 

WNES    AND    TAN'GEN'TtJ.       (31     DEGREKt5.) 


41) 


W. 


9 
in 
II 
13 
|3 
14 
|5 
i6 

\l 

19 

20 
21 
22 
23 
24 
25 
26 

II 

3i 

32 

33 
34 
35 

36  i 

ll 
3q 
40 
41 
42 
43 
44 
45 
46 

% 

5o 
5i 

52 

53 
54 
55 
56 

U 


Sine 


9-711839 
7i2o5o 
712260 
712469 

712880 
713098 
7i33o8 
713517 
713726 
713935 
9-7I4I44 
714352 
71456! 
714760 
714978 
7i5t86 
7 1 5391 
7 1 56o2 
7 1 5809 

7 1 60 1 7 ! 
9-716224 
716432 
716639! 
716846! 
7170531 
717259' 
717466. 
717673. 
717879 

7i8o85 
9-718291' 
718497 
718703 
718909; 
7i9"4i 
719320 
719525 
719730 
719935 
72or4o 
9-720345 
750549' 
7207J4 
720953 
721162' 
721366' 
72i57c| 
721774' 
721978 
722181I 
9-722385 
722588 
722791! 
722994 
723197, 
7 23400 I 
7236o3| 
7238o5 
724007 
724210 


D. 


Cosine    I  D.  |    Taii^.    i      D.      |  Cotaiig.  f~ 


3-50 
3-50 
3-50 
3-49 
3-49 
3-49 
3.49 
3.4$ 
3.4b 
3.48 
3-48 


3-45 

3-45 

3.45 

3-45 

3-45 

3-44 

3-44 

3-44 

3-44 

3.43 

3-43 

3.43 

3-43 

3.43 

3-42 

3-42 

3.42 

3.42 

3.41 

3.41 

3 

3 

3 

3 

3- 


41 
40 
4c 
40 

3-40 

3.40 

3 

3 

3 

3 

3 

3. 


CJosine 


39 
39 

ll 

It 

3-38 
3-38 
3-38 
3.37 
3.37 
3.37 
3.37 


9-933o66'i-26| 
982990  I -27! 
932oi4;i-27| 
932$;j8  1-271 
932]62  i-27| 
932685; I -271 
93260Q  1-27! 
932534  1-27I 
932457  I -271 
932380  1-27! 
982304  I  -27, 

9932228  1-27' 
932i5i  1-27' 
982075  I -28! 
981998  1-28, 
981 921' I -281 
981845  I-28| 
981768  1-28^ 
981691  1-28' 
981614  1-28, 
981587I1.28 

9>93i46o  I  -28 
981883  1.28j 
981806  1-28 
981229  1  -29' 
981 1 52  I  -29' 
981075  1-29: 
980998  1-29' 
93oo2i[i.29l 
980843, 1 -29 
980766  1-29' 

9-980688  1-29' 
98061 1  I  -291 
98o538  1-29^ 
980456  I  -291 
9808781 1.29I 
980800  i.3oj 
980228  I -3oi 
980145  1  -30] 
980067  1 .3o; 
92998Q  1 .3o| 

9-92091 1  i-3oi 
929888, 1. 3oj 
929755  I -80 
929677  i.3o 
929599  1 .30 
929531  I -801 
929442  i.3o 
929 ?6^  i-8ij 
9292%  i-3i' 
929207  1 .31 ; 

9-929129  I  -31 
92905c  1-81 
9:8972  I  -311 
928K98  I  81 
928815  i.3r 
928786  1.31 1 
928657  I  •  3 1 
92S5781.81 
Q28499  I  -31 
I       ^284201 -31  i 


D. 


I680 


9-778774; 
779060 
779346 
779682I 
7799'8{ 
780203 
780489 
780773; 
7810^)0. 
781846I 
781681 

9-781916 
782301, 
7S2486 
782771 
7«8o56 
7888  J I 
788626 
788910 
784195 
784479 

9-784764 
78,^50 ',8 
785882 
7856 1 6 
785900 
786181 
786468 
■786752 
787086 
787810 

9-787608 
7878S6 
78S170 
788453 
788786 
789019 
789I02 
789585 
789868 
790i5i 

9-790488 
790716 
790999 
7912S1 
791568 
791846 
792128 
792iio 
792692 
79297 i 

?• 798256 
79^538 
7938 1 9 
79iioi 
794883 

79^045 
795327 
795508 
i   795789 
1^  Cot&n^, 


4-77 
4- 


4-77 
4-76 
76 
76 
76 
76 
76 
76 
75 
75 
75 
75 
75 
75 


4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4-7-' 

4-75 

4-74 
74 
74 
74 
74 
74 
73 
73 
73 
73 
73 
73 
73 
72 
72 
72 
72 
72 
72 
72 
7> 
7' 


4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4' 

4 

4-7« 

4-71 


469 
469 
4  69 
4  69 
4-69 
4-69 
4-69 
4-68 
4-68 


I|0-22I226 

220940 
220634 
220368 
220082 

219797 

2I95I I 
219225: 

2  I K940 

218654) 

2IS869I 

10  218084 

2i779<; 

217314 

217229 

2  !  69 
216659 
216874 
2  1 6090 

2i58o5 
2 1 552 1 

10-215286 
214952 
214668 
214884 
2 1 4 1 00 
218816 
218582 
218248 
2 1 2964 
212681 

10-212897 
212114 
211880 
21 1 547 
2 1 1 264 
2 1 098 1 
2 1 0698 
2 1 041 5 
210182 
209H49 

10-209567 
209384 
209001 
208719 
208487 
208 1 541 
207872 
207590 
2,t78o8 
207026] 

10-206744 
206462 
206181 
205899 
205617I 
2o5336l 
2o5o55' 
204773 
204492 
2042 1 1 


60 

u 

55 
54 
53 

52 

5i 

5o 

47 
46 
45 
44 
43 
42 
41 
40 

ll 

35 
34 
33 

32 

3i 
3o 

It 
ll 

25 

24 

23 
22 
21 
20 


I  Taiiff. 


50 

(32  DEGREES.)   A 

TABLE  OP  LOGARITHMIC 

M. 

0 

Sine 

1   ^• 

Cosine  |  D. 

Tanof. 

1   ^' 

Cotaiig.  1 

9-724210 

3-37 

9-928420  1-82 

9.795789!  4-68 

10' 20421 1 ;  60 

I 

724412 

1  3 

•37 

928842  1-32 

796010 

4.68 

2089301  5o 

2 

724614 

1  3 

•36 

928268  I  -82 

79633! 

4-68 

2086491  58 
203368!  57 
208087!  56 

3 

724816 

3 

-86 

928188 

1-82 

796682 

4-68 

4 

725017 

3 

-86 

928104 

1-32 

796913 

4-68 

5 

725219 

3 

•  36 

928025  1-82 

797194 

4-68 

202806!  55 

6 

725420 

3 

35 

927946  1-82 
927867  1-82 

797475 

4-68 

20x525;  54 

2 

725622 

3 

85 

797755 

4-68 

202245!  53 

725823 

3 

35 

927781  1-82 

798086 

4-67 

20.964;  52 

9 

726024 

3 

35 

927708  1-32 

798816 

4-67 

2oi684j  5! 

10 

726225 

3 

35 

927629  1-82 

798596 

4-67 

201 404 j  5o 

II 

9.726426 

3 

34 

9.927549  1-32 

9.798877 

4-67 

IO-20I128l  49 

2008431  48 

la 

726626 

3 

34 

927470  1-38 

799157 

4-67 

i3 

726827 

3 

34 

927890 

1.38 

799437 

4-67 

2oo563  47 

U 

727027 

3 

34 

927810 

1.83 

799717 

4-67 

200283  46 

i5 

727228 

3 

34 

927281 

!.33 

799997 
800277 

4-66 

200008  45 

i6 

727428 

3 

33 

927.51 

1.83 

4-66 

199723'  44 

\l 

727628 

3 

33 

927071 

1.88 

800557 
800886 

4-66 

199448!  43 

727828 

3 

38 

926991 

1.88 

4-66 

199164  42 

^9 

728027 

3 

33 

9269TI 
926S81 

!.33 

801 1 16 

4-66 

19S884!  41 

20 

728227 

3 

33 

1.33 

801896 

4-66 

198604!  40 

21 

9-728427 
728626 

3 

32 

9-926-51 

1.38 

9.801675 
801955 

4-66 

10-198825;  39 

198045;  38 

22 

3 

82 

026671 

1.38 

4-66 

23 

728825 

3 

32 

926591 

!-33 

802284 

4-65 

197766'  37 

24 

729024 

3 

82 

9265 11 

1.34 

8o25i3 

4-65 

197487;  36 

25 

729223 

3 

81 

926481 

1-84 

802792 

4-65 

197208  35 

26 

729422 

3 

81 

926851 

1-34 

808072 

4-65 

196928  34 

11 

-720621 

3 

81 

926270 

1-34 

8o885i 

4-65 

196649  33 

729820 

3 

3i 

926190 

1-34 

808680 

4-65 

196870;  32 

29 

780018 

3 

3o 

926110 

1-84 

808908 

4-65 

196092  3 1 

3o 

780216 

3 

3o 

926029 

1-34 

804187 
9-804466 

4-65 

195813,  3o 

3i 

9-73o4i5 

3 

80 

'■fM^ 

1-34 

4-64 

10- 195534  29 
1952551  28 

32 

7806 1 3 

3 

3o 

1-34 

804745 

4-64 

33 

780811 

3 

3o 

925788 

1-34 

8o5o23 

4-64 

I 949771  27 
1946981  26 

34 

781009 

3 

29 

925707 

1-34 

8o53o2 

4-64 

35 

73 1 206 

3- 

29 

925626 

1-34 

8o558o 

4-64 

194420  25 

36 

781404 

3. 

29 

925545 

1-35 

8o5859 

4-64 

I94i4!l  24 

ll 

781602 

3- 

29 

925465 

1-35 

806187 

4-64 

198868!  23 

781799 

3- 

U 

925884' 

1-35 

8064 1 5 

4-63 

193585,  22 

39 

781996 

3- 

925308: 

1-35 

806693 

4-63 

198807 

31 

40 

782.98 

3. 

28 

925222I 

!-85 

806971 

4-63 

198029 

20 

41 

9-732390 

3. 

28 

9-925i4i 

1.35 

9-807249 

4-68 

.0-192751 

\t 

42 

782587 

3- 

28 

925o60| 

1.35 

807527 

4-63 

192473 

43 

782784 

3- 

28 

924979, 

1.35 

807805 

4-63 

,  192.95 

ll 

44 

782980 

3- 

27 

924897 

!.85 

80S088 

4-68 

191917 

45 

788.77 

3- 

27 

924816 

1-85 

808861 

4-63 

.9.689 

:5 

46 

783378 

3. 

27 

924735 

1-86 

808688 

4-62 

191862!  14 

ii 

783569 
788765 

3- 

27 

924654; 

1-86 

808916 

4-62 

.91084  18 

3. 

27 

924572J1-86 

809.98 

4-62 

190S07J  12 

49 

7330. 

3. 

26 

924491; I -36 

809471 

4-62 

190329;  11 

5c 

784157 

8. 

26 

924409' 1  -86 
9-924828, 1-36 

809748 

4-62 

190252!  !0 

5i 

9-784553 

8 

26 

9.810025 

4-62 

10-1899751   Q 

1896981  8 

§2 

784549 

3- 

26 

924246  1-36 

810802 

4-62 

53 

734744 

3- 

2: 

924i64:i-36 

8io58o 

4-62 

189420   7 

54 

734989 

8- 

2) 

924088  I -361 

810857 

4-62 

189.43 

188866 

6 

55 

785.85 

3- 

25 

924001;!  '36 

811184 

4-6i 

5 

56 

78588o 

3- 

25 

9289 1 9' 1-36.' 

811410 

4.61 

188590 

4 

ll 

785525 

3- 

25 

928887, 1 -36; 

811687 

4-6i 

i883i3 

3 

735719 

-i- 

24 

928755;!. 87 

81 1964 

4-61 

i88o36 

a 

59 

735914 

3. 

24 

928678  1-37 
92359IJI-37 

812241 

4-61 

»?775q  I  1 

6o 

786109 

3-24 

812517 

4-61 

187483 

0 

Cosine 

D. 

Sine  J5TO! 

Cotang.  i 

D.   1 

Tang. 

M. 

BINBG 

A  NT)  TANGENTS. 

(3.3  DKORRKS. 

) 

b\ 

\^ 

bine 

D. 

Cosine  |  D.  |  Tamr. 

D. 

Cot&ng. 

1 

o 

'•]& 

3.24 

9-92359i|i.37i  9.812517 

61 

10-187482 

"60 

I 

3 

24 

923509' I 

•371   8!2794 

61 

187206 

59 

a 

736498 

3 

24 

923427;! 
923J45!i 

•37 

813070 

61 

186930!  58  1 

3 

?a? 

3 

23 

•37 

813347 

4 

.60 

186653 

57 

4 

3 

23 

923263  I 

•37 

81 3623 

4 

60 

186377 

5o 

5 

737080 

3 

23 

923i8i]i 

•37 

813899 

4 

60 

186101 

55 

6 

737374 

3 

23 

923098' I 

.37'   814175 
.37!   8!4452 

60 

185825 

54 

I 

737467 

3 

23 

923oi6|i 

60 

185548;  53 

737661 

3 

22 

922q33i 
92285i  1 

.37 

814728 

60 

.85272  5j 

g 

737855 

3 

22 

•37 

8i5oo4 

60 

184996  5i  1 

IC 

738048 

3 

22 

9227681 
9 .922686 1 1 

.38 

8i5279 

60 

18472 1  5o  ^ 

II 

9-738241 

3 

22 

.381  o.8i5555 

59 

10-184445  49 
.84169  4!^ 
183893'  i7 

19 

738434 

3 

22 

922603  I 

-38!   8i583i 

59 

!3 

738627 

8 

21 

92252o'l 

-381   8r6i07 

59 

t/i 

738820 

3 

21 

922438,1 

-38i   8i6382 

59 

1 836 18]  46 
183342  45 

i5 

739013 

3 

21 

922355  I 

-38 

8 16658 

59 

|6 

739206 

3 

21 

922272  I 
922189  I 

•  38 

816933 

59 

1 83067 1  44 

]l 

739398 

3 

21 

.38 

817209 

^ 

182791!  43 

739500 

739783 

3 

20 

922106  I 

-38 

817484 

59 

182516  42 

«9 

3 

20 

922023  I 

.38 

8.7759 
8i8o35 

U 

182241!  41 

2o 

739975 

3 

20 

921940  I 
9-921857  I 

-38 

181965 
10-181690 

40 

21 

9-740167 

3 

20 

.39  9-8i«3io 

58 

^ 

22 

740359 

3 

20 

931774  I 

.39 

8.8585 

58 

181415 

23 

74o55o 

3 

19 

92169I  I 

.39 

818860 

53 

181140J  37 

24 

740742 

3 

19 

921607  I 

-39 

819135 

58 

.8o865i  36 

25 

740934 

3 

19 

921524  I 

-39 

819410 

58 

.R0590!  35 

26 

741125 

3 

»9 

921441  1 

-39 

819684 

58 

1803 i6j  34 

27 

74i3i6 

3 

\t 

921357  I 

.39 

8.0959 

58 

18004 1 1  33 

28 

74i5o8 

3 

9312741 

-39!   820234 

58 

1797661  32 

2y 

741699 

3 

18 

921 190  I 

.39   82o5o8 

4 

57 

1794931  3i 

3o 

741889 

3 

18 

921 107  I 

.39 

820783 

/ 

57 

170217'  3o 

3i 

9.742080 

3 

18 

9-921023  I 

-39 

9-821057 

57 

io-ii>^,,iZ    29 
178668;  38 

32 

743271 

3 

18 

Pl?! 

•40 

82.332 

57 

33 

742462 

3 

17 

.40 

821606 

4 

57 

178394;  37 

34 

742652 

3 

17 

920772'! 

.40 

821880 

, 

57 

1 78 120,  26 

35 

743842 

3 

17 

920688! ! 

.40 

822154 

57  , 

177846  25 

36 

743o33 

3 

17 

920604!! 

.40 

822429 

57  ^ 

177571;  34 

37 

743223 

3 

'7 

92o52o!i 

.40 

822703 

ll 

177297!  23 

38 

743413 

3 

16 

920436|i 

.40 

822977 

177023i  22 

39 

743602 

3 

16 

920352I1 

.40   823250 

56 

17675OJ  2. 

40 

9-743982 

3 

16 

920268,1 

-40!   823524 

56 

.76476,  20 

.  ii 

3 

16 

9-930i84li 

-40;  9-823798 

56 

10.176202 

\l 

42 

744171 

3 

16 

920099  I 

.40   824072 

56 

175928 

175655 

43 

744361 

3 

i5 

9300l5|I 

•40!   824345 

56 

17 

44 

744550 

3 

i5 

919931  I 
919846  I 

•41   824610 
.41   82489J 

4 

56 

175381 

16 

45 

744730 
74493S 

3 

i5 

56 

175.07  15 

46 

3 

i5 

919762  I 

.41   825166 

56 

174834  14 

S 

745117 

3 

i5 

9J9677  I 

-41   825i3Q 
■41   825713 

55 

174561  13 

745306 

3 

14 

919593  I 

55 

174287  !2 

49 

745404 

3 

14 

919508,1 

•41 

825986 

55 

174014  «I 

5u 

745683 

3 

14 

919434  1 

•41 

826259 

55 

1737411  10 

5i 

9-745871 
746o5o 
74624B 

3 

14 

9919339  I 

•41 

9-826532 

55 

10-173468   9 

92 

3 

14 

9192541 

•41 

826805 

55 

173195  8 

« 

3 

i3 

919169!! 

919085:1 

-41 

827078 

55 

172922  7 
1726491  6 

54 

746436 

4 

i3 

.41 

8273DI 

55 

55 

746624 

3 

i3 

919000'! 

•41 

827624 

55 

172376]  5 

56 

746812 

3 

i3 

9i89!5:! 

•42 

827897 
828170 

54 

172103   4 

■u 

746909 
747 '87 

3 

i3 

9i883o  ! 

•43 

54 

171830  3 

3 

12 

918745  I 
918659'. 

-42 

828442 

54 

I7I558   2 

^) 

747374 

3 

12 

-42 

828715 

54 

171285   I 

6o 

747562 

3  12 

918574  ! 

•42 

828987 

54 

171013  0 

_Coamo_ 

D. 

Sine   6O0I 

Cotang^ 

~D.~I 

.  Tanff. 

^J 

52 

(34  DEOREEB.)   A 

TABLE  OF  LOGARITHMIC 

M. 

Bine 

D. 

Cosine  1  D. 

1  Tang. 

D. 

Cotang. 

1    ' 

o 

9 -747562 

3.12 

9.9i8574;i-42 
918489' 1. 42 

9-828987 

4-54 

10-171013 

1  60 

I 

747749 

3.13 

829260 

4-54 

170740 

a 

747936 

3-13 

918404  1-42 

829532 

4-54 

170468 

58 

3 

748123 

3. II 

9i83i8|i.42 

829805 

4-54 

170195 

57 

4 

748310 

3.11 

918233I1.42 

830077 

4-54 

169923 

56 

5 

748497 
7486$3 

3. II 

918147 

1-42 

83o349 

4.53 

I 6965 I 

55 

6 

3-11 

918062 

1.42 

83o62i 

4-53 

169879 

54 

I 

748870 

3.11 

917976 
917891 

1-43 

1   830893 

4.53 

16^835 

53 

749056 

3.10 

1.43 

83ii65 

4-53 

5a 

9 

749243 

3.10 

917805 

1.43 

831437 

4-53 

168563 

5i 

10 

749429 
9-749615 

3.10 

917719 

|i-43 

881709 

4-53 

168291 

5o 

II 

3.10 

9-917634 

|i-43 

9.831981 

4-53 

10.168019 

^ 

la 

749801 

3-10 

917548  1.43 

832253 

4-53 

167747 

i3 

749987 

3.09 

917462  1.43 

832525 

.4-53 

167475 

2 

a 

750172 

3.09 

917376  1.43 

832796 

4-53 

167204 

i5 

75ol')8 

3.09 

91729011.43 

833o68 

4-52 

i6'j932 

45 

i6 

75o543 

3-09 

917204 

1.43 

833339 

4-52 

166661 

44 

\l 

700729 

3':^^ 

917118 

1.44 

833611 

4-52 

166889 
166118 

43 

7D0914 

917032 

1.44 

833882 

4-52 

43 

J9 

Villi 
9.751469 

3-08 

916046 

1-44 

834 1 54 

4-52 

165846 

41 

20 

3-08 

916859  1.44 
9.91677311.44 

834425 

4-52 

165375 

40 

31 

3.08 

9.834696 

4-52 

io-i658o4 

^ 

23 

75i654 

3.08 

916687 

1.44 

834967 

4-52 

i65o83 

23 

75i83q 

732023 

3.08 

916600 

1-44 

833238 

4-52 

164762 

37 

24 

3.07 

9i65i4 

1.44 

835509 

4-52 

164491 

36 

23 

732208 

3-07 

9 -'642  7 

1-44 

835780 

4-5i 

164220 

35 

26 

732392 

3.07 

916341 

1-44 

836o5i 

4-5i 

168949 

34 

u 

752576 

3.07 

916254 

1-44 

836322 

4-5i 

168678 

33 

752760 

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39 

785925 

73 

898592 

1-62 

887888 

4-35 

21 

40 

7860H9 

73 

898494 

1-63 

887594 

4-35 

112406 

20 

41 

9-7S6252 

72 

9-898897 

1-68 

'•XI 

4-35 

10-112145 

\t 

42 

786416 

72 

898299 

1-63 

4.35 

1 11884 

43 

786579 

72 

898202 

1-63 

888877 

4-35 

111628 

\l 

44 

786742 

72 

898104 

1-63 

888689 

4.35 

Iii36i 

45 

786906 

72 

898006 

1-63 

888900 

4-35 

lillOO 

i5 

46 

787069 

72 

897008 

897810 

1-68 

889160 

4-35 

110840 

14 

s 

787282 

71 

1-68 

880421 

4-35 

110818 

i3 

787895 
787557 

71 

897712 

1-68   889682 

4-35 

12 

49 

7« 

891614 

1-63   889948 

4.35 

110057 

II 

5o 

787720 

71 

897516 

i-68|   890204 

4-34 

109796 

10 

5i 

9-787888 

V 

9-897418 

1-64'  9  890465 

4.34 

10-109535 

I 

52 

788045 

V 

897820 

i-64i   890725 

4-34 

109275 

53 

788208 

71 

897222 

1-64!   890986 

4-34 

109014 
10S753 

7 

54 

]mi 

70 

gi 

1-64;   891247 

4-34 

6 

55 

70 

1.64   891507 

4.34 

108233 

5 

56 

7S 

70 

1-64!   891768 

4-34 

4 

ll 

70 

i-64i   892028 

4-34 

10-.972 

3 

789018 

70 

896729 

I -641   892289 

4-34 

107711 

3 

59 

789180 

70 

896681 

1-64   892549 

4-34 

107451 

I 

60 

789342 

69 

896582 

I -641   892810 

4-34 

107190 

0 



Coeiro 

D. 

Sine 

e20|  Cot!in£r. 

D. 

Tiinj?. 

M. 

27' 


56 


(38    DEGREES.)       A    TABLE    OF    LOGARITHMIC 


pr 

Sine 

D. 

Cosine 

I). 

Tang. 

D. 

Cctang. 

'   ~ 

0 

9-789342 

2.69 

"9^896532 

1-64 

"^8928lc 

4-34 

10-107190 
106930 

60 

I 

789504 

2.69 

806433! I -65 

89307c 

4.34 

5? 

3 

789665 

2.69 

896335 

11-65 

893331 

4-84 

106669 

3 

789827 

2-69 

896286 

1-65 

893591 
898851 

4-34 

106409J  57 

4 

•789988 

2-69 

896187 
896088 

1-65 

4.34 

106 149;  56 

5 

790 '49 

1-65 

8941  u 

4-34 

109889 

55 

6 

790810 

895089 

1-65 

894871 

4.34 

ioj62c 
105368 

54 

I 

790471 

2-68 

895840 

'i-65 

894682 

4-38 

53 

790682 

2-68 

895741 

1-65 

894892 

4-33 

io5io8 

52 

9 

790798 

2-68 

895641 

1-65 

895152 

4-83 

104848 

5i 

10 

790934 

2-68 

895542 

1-65 

895412 

4-33 

104588 

5o 

II 

9-791II5 

2-68 

9-895448 

1-66 

9-895672 

4-38 

10-104328 

i^ 

12 

791275 

2.67 

895848 

1-66 

890982 

4-33 

104068 

i3 

791486 

2-67 

895244 

1-66 

896192 

4-83 

io38o8 

47 
46 

14 

791596 

2.67 

895145 

1-66 

896402 

4-33 

103548 

i5 

791737 

2.67 

895045 

1-66 

896712 
8^6971 

4-83 

108288 

45 

i6 

791917 

2.67 

894945 
894846 

1-66 

4-83 

108029 

44 

\l 

792077 

l:tl 

1-66 

897281 

4-83 

102769 

43 

792287 

894746 

1-66 

897491 

897751 

4-83 

102509 

42 

19 

792897 

2-66 

894646 

1-66 

4-33 

102249 

41 

20 

792537 

2-66 

894546 

1-66 

898010 

4-33 

101900 
10-101780 

40 

21 

9-792716 

2-66 

9-894446 

1.67 

9-898270 

4-33 

^? 

22 

792876 

2-66 

894846 

1.67 

898530 

4-88 

101470 

38 

23 

798085 

2-66 

894246 

1.67 

898789 

4-38 

101211 

37 

24 

]llt 

2-65 

894146 

1-67 

899049 
899808 

4-32 

100951 

36 

25 

2-65 

894046 

1.67 

4-32 

100692 

35 

26 

79^514 

2-65 

898046 
898846 

1-67 

899568 

4-82 

100482 

34 

11 

798673 

2-65 

1-67 

899827 

4-32 

100178 

33 

798882 

2-65 

898745 

1-67 

900086 

4-32 

099914 

32 

^9 

798991 

2-65 

898645 

1-67 

900846 

4-82 

099654 

3i 

3o 

794 I 5o 

2-64 

898544 

1.67 

900605 

4-32 

099805 

3o 

3i 

9-794808 

2-64 

9-898444  1-68 

9-900864 

4-32 

10-099186 

29 

28 

32 

794467 

2-64 

898848 

1-68 

901124 

4-32 

09S876 

33 

794626 

2-64 

898243 

1-68 

901888 

4-32 

098617 

11 

34 

794784 

2-64 

898142 

1-68 

901642 

4-82 

098858 

35 

794942 

2-64 

898041 

1-68 

901901 

4-82 

098099 

25 

36 

795101 

2-64 

892940 

1.68 

902160 

4-82 

097840  24  1 

11 

795259 

2-63 

892889  1-68 

902419 

4-32 

097581 

23 

795417 

795575 

795733 

9-795891 

2-63 

892739:1.68 
89268811-68 

902679 
902988 

4-32 

097821 

22 

39 

2-63 

4-82 

097062 

21 

40 

2-63 

892586{i.68 

9-908455 

4-3i 

096808 

20 

41 

2-63 

9-892435:1.69 

4-3i 

10-096545 

13 

42 

796049 
796206 

2-63 

8923341-69 

908714 

4-3i 

096286 

43 

2-63 

892233!  I- 69 

908978 

4.31 

096027 

17 

44 

796864 

2-62 

892182  1.69 

904282 

4-3i 

095768 

16 

45 

796521 

2-62 

892080! I -69 

904491 
904750 

4-3i 

095509 

i5 

46 

796679 

2-62 

891929' 1-69 

4-3i 

095250 

14 

% 

49 

796886 

2-62 

8918271-69 

9o5oc8 

4-3i  ! 

094992 
OV74733 
094474 

i3 

796993 
797 1 5o 

2-62 

2-61 

891726I1-69 
891624' I -69 

905267 
905526 

4-8i  1 
4-3i 

12 

5o 

797807 

2-61 

891528  1.70 

905784 

4-3i 

094216 

10 

5i 

9-797464 

2-61 

9-891421 

1-70 

9-906048 

4-3i 

10-093957 

t 

52 

797621 

2-61 

891819 

1-70 

90680a 

4-3i 

098698 

53 

797777 
797984 

2-6l 

891217 

1-70 

9o656o 

4-3i 

098440 

I 

54 

2-61 

891 1 15 

1-70 

906819 

4-3i 

098 181 

55 

798091 

2.61 

891018 

1-70 

907077 

4-3i 

092Q23 

5 

56 

798247 

2.61 

89X9 

y-70 

907886 

4  3i 

092664 

4 

U 

798403 

2-60 

1-70 

907594 
907852 
9081 1 1 

4-3i 

092406 

3 

798560 

2.60 

890707:1.70 

4-31 

092148 

a 

59 

« 

2.60 

890605^1-70 

4-30 

091889 

I 

6o 

2.60 

890508  1.70 

908869 

4-3o 

091681 

0 

CoBine 

D. 

Sine  la  10 

Cotang, 

D. 

Tang. 

SINES 

AND  TANGENTS. 

(39  DEGREES. 

) 

61 

IT. 

0 

Sine 

D. 

Cosine  |  D. 

Tauor. 

D. 

Cotang.  1    1 

9.798872 

2-60 

9.890503:1.70 

9-9o83tQ 

4.30 

10-091631 

60 

I 

799028 

60 

890400  1. 7 1 

90862S 

3o 

091372 

^ 

a 

799184 

60 

890298  1. 7 1 

908886 

3o 

091114 

58 

3 

799339 

59 

890195  1.71 

909144 

3o 

090856 

57 

4 

799493 

59 

Sl;:^; 

909402 

3o 

090598  56  1 

5 

799631 

5? 

009660 

3o 

090340 

55 

6 

799806 

59 

9099 1 8 

3o 

090082 
089823 

54 

I 

•799962 

59 

889785  I.? I 

910177 
910435 

3o 

53 

8001 17 

59 

889682:1.71 

3o 

089565 

52 

9 

800272 

58 

889579>-7» 

910693 

3o 

089307!  5l 

10 

800427 

38 

88g477ii-7i 

9i09-)i 

3o 

089049,  5o 

IX 

9.800582 

58 

9-889374  1.72 

9.911209 

3o 

10-088791 
088533 

it 

la 

800737 

58 

8892711.72 

91 146- 

3o 

1  '3 

800892 

2 

58 

8891681.72 

911724 

3o 

088276 

47 

14 

801047 

58 

iEPi 

9119S2 

3o 

088018 

46 

i5 

801201 

58 

912240 

4 

3o 

087760 

45 

i6 

80 1 356 

2 

57 

912498 

4 

3o 

087502 

44 

\l 

8oi5ii 

57 

88875511.72 

912756 

4 

3o 

087244 

43 

80 1 665 

57 

88S651I1.72 

9i3oi4 

4 

29 

086086 

42 

«9 

801819 

57 

888548!  1. 72 

913271 

4 

29 

086729 

41 

20 

801973 

57 

888444I1.73 

913529 

4 

29 

086471 

40 

31 

9-802128 

57 

9-88834i;i.73 

9-9I3787 

4 

29 

10-086213 

It 

72 

802282 

56 

8882371.73 

914044 

4 

29 

085956 

23 

802436 

56 

8881341.73 

914302 

4 

29 

085698 

u 

24 

802589 
802743 

56 

8880301.73 

914560 

4 

29 

085440 

25 

56 

8879261.73 
8878221.73 

914817 

4 

29 

o85i83 

35 

26 

802897 

56 

9i5o75 

4 

29 

084925 

34 

U 

8o3oDo 

56 

887718I1.73 

915332 

4 

29 

084668 

33 

803204 

56 

8876141.73 

915590 

4 

29 

084410 

32 

29 

803357 

55 

8875io'i.73 

9'5847 

4 

29 

084 1 53 i  3 1 

3o 

8o35ii 

55 

887406' 1-74 

916104 

4 

29 

083896  3o 

3i 

9.803664 

55 

9-887302,1-74 

9-916362 

4 

29 

10-083638 

It 

32 

8o38i7 

55 

887198!  I-. 74 

916619 

4 

29 

08338 1 

33 

803970 

55 

886885!  1 .74 

916877 

4 

29 

o83i23 

27 

34 

804123 

55 

9<7>34 

4 

29 

082866 

26 

35 

804276 

34 

917391 

4 

29 

082609 

25 

36 

804428 

54 

886780  1-74 

917648 

4 

29 

082352 

24 

ll 

804581 

54 

886676  1-74 

917905 

4 

It 

082095 
081837 

23 

804734 
804886 

54 

886571  1-74 

918163 

4 

22 

39 

54 

8864661.74 

918420 

4 

28 

o8i58o 

21 

4o 

8o5o39 

54 

8863621 1. 75 

918677 

4 

28 

08 1 323 

20 

4i 

Q-8o5i9i 

54 

9. 886257 '1.75 

9-918934 

4 

28 

10-081066 

\t 

42 

805343 

53 

886152:1.75 

919191 

4 

28 

080809 

43 

805495 

53 

886047!  1. 75 

919448 

4 

28 

o8o552 

]l 

44 

8o5647 

53 

885042^1-75 
885837  1.75 

919705 

4 

28 

080295 

45 

805709 
80595 1 

53 

919962 

4 

28 

o8oo38  1 5  1 

46 

53 

8857321.75 

920210 
920476 

4 

28 

079781 

14 

% 

806 1 o3 

53 

885627  1.75 

4 

28 

079524 

i3 

806254 

53 

885522  1.75 

920733 

4 

28 

079267 

12 

49 

806406 

52 

8854161.75 

920990 

4 

28 

0790 10  II 
078753  10 

5o 

806557 

52 

88531111.76 

921247 
9-92i5o3 

4 

28 

5i 

'■»o 

52 

9-8852o5li.76 

4 

28 

10-078497:  9 
078240I  s 

52 

52 

885 100' 1.76 

921760 

4 

28 

53 

807JII 

52 

884004:1-76 
8848ao'i.76 
884783:1.76 

Q22017 

4 

28 

0779831  7 
077726.  6 

54 

807163 

52 

922274 

4 

28 

55 

807314 

52 

922530 

4 

28 

077470 

5 

56 

807465 

5i 

8846771.76 

922187 

4 

28 

C72956 

U 

807615 

5i 

8845721.76 

923044 

4 

28 

3 

807766 

5i 

8844661.76 

923300 

4 

28 

076700 

i 

te? 

5i 
5i 

8843601.76 
884254  1-77 

9238i3 

4 
4 

27 
27 

076443 
076187 

Twiff.- 

-M-. 

Cosino 

D. 

Sine  IfiQo 

Cotem?. 

'Z? 

>.   1 

5tt 

(40 

DEGREES.)   A 

TABLE  OF  LOGARITHMIC 

M. 

Sine 

D. 

Cosine 

D. 

Tung. 

D. 

Cotan^. 

0 

0-808067 

2.5l 

9.884354 

9 -92381 3 

4 

•27 

10-076187 

60 

I 

808318 

2 

5i 

884148 

924070 

4 

.27 

075980 

5? 

3 

8o8368 

2 

5i 

884042 

924327 

4 

27 

075673 

3 

8o85i9 

2 

5o 

883936 

924583 

4 

27 

075417 

U 

4 

808669 

2 

5o 

883829 
883723 

924840 

4 

27 

075160 

5 

808819 

2 

5o 

925096 

4 

27 

074904 

55 

6 

808969 

2 

5o 

8836 1 7|  I 

925332 

4 

27 

074648 

54 

I 

8091 19 
809269 

2 
2 

5o 
5o 

883510 
883404 

]] 

92586? 

4 
4 

27 
27 

074891 
074135 

53 

52 

9 

809419 

2 

49 

883297 

78 

926122 

4 

27 

078878 

5i 

10 

809569 
9.809718 

2 

49 

883191 
9.883084 

78 

926378 

4 

27 

078622 

5o 

II 

2 

49 

78 

9.926634 

4 

27 

ic- 078866 

% 

13 

809868 

2 

49 

882977 

78 

926890 

4 

27 

078110 

i3 

810017 

2 

49 

882871 

78 

927147 

4 

27 

072853 

.% 

14 

810167 

2 

'S 

882764 

78 

927403 

4 

27 

072597 

i5 

8io3i6 

2 

882657 

78 

927659 
9271915 

4 

27 

072841 

45 

i6 

810465 

2 

48 

882550 

78 

4 

27 

072085 

44 

\l 

810614 

2 

48 

882443 

78 

928171 

4 

27 

071829 

071573 

43 

810763 

2 

48 

882336 

79 

928427 

4 

27 

42 

J9 

810912 

2 

48 

882229 

79 

928683 

4 

27 

071817 

41 

20 

811061 

2 

48 

882121 

79 

928940 

4 

27 

071060 

40 

21 

9-8ii2io 

2 

48 

9-882014 

79 

9.929196 
929452 

4 

27 

10  070804 

39 

38 

23 

8ii358 

2 

47 

881907 

79 

4 

27 

070548 

23 

8ii5o7 

2 

47 

881799 

79 

929708 

4 

27 

070292 

37 

24 

8ii655 

2 

47 

881692 

79 

929964 

4 

26 

070086 

36 

25 

81 1804 

2 

47 

88 1 584 

79 

980220 

4 

26 

069780 

35 

26 

811952 

2 

47 

881477 

79 

930475 

4 

26 

069525 

34 

27 

812100 

2 

47 

881369 

79 

930781 

4 

26 

069269 

33 

28 

812248 

2 

47 

881261 

80 

980987 

4 

26 

069013 

32 

29 

812896 

2 

46 

881 1 53 

80 

981248 

4 

26 

068757 

3i 

3o 

812544 

2 

46 

881046 

80 

981499 
9.931755 

4 

26 

o685oi 

3o 

3i 

9-812692 

2 

46 

9-880938 

80 

4 

26 

10-068245 

ll 

33 

812840 

2 

46 

88o83o 

80 

982010 

4 

26 

067990 

33 

812988 

2 

46 

880722 

80 

982266 

4 

26 

067784 

27 

34 

8i3i35 

2 

46 

880613 

80 

982522 

4 

26 

067478 

26 

35 

8i3283 

2 

46 

88o5o5ii 

80 

982778 

4 

26 

067222 

25 

36 

8i343o 

2 

45 

880397  I 

80 

988088 

4 

26 

066967 

24 

ll 

813578 

2 

45 

880289 

I 

81 

988289 
988545 

4 

26 

066711 

23 

813725 

2 

45 

880180 

I 

81 

4 

26 

066455 

22 

39 

813872 

2 

45 

880072!  I 

81 

988800 

4 

26 

066200 

31 

40 

814019 

2 

45 

879Q63I 
9-879855!! 

81 

984056 

4 

26 

063944 

ao 

41 

9'8i4r66 

2 

45 

81 

9-984811 

4 

26 

10-065689 

\i 

42 

8i43i3 

2 

45 

879746:1 

81 

984567 

4 

26 

065433 

43 

814460 

2 

44 

879637' I 

81 

984828 

4 

26 

o65i77 

u 

44 

814607 

2 

44 

879529  I 

81 

935078 

4 

26 

064922 

45 

814753 

2 

44 

879420 

I 

81 

935383 

4 

26 

064667 

i5 

46 

814900 

2 

44 

879311 

I 

81 

985589 

4 

26 

064411 

14 

% 

8 1 5046 

2 

44 

879202  I 

82 

935844 

4 

26 

064 1 56 

i3 

815193 
815339 

2 

44 

870003!  i 
8780841 1 
878§75|i 

82 

9861  .■>o 

4 

26 

068900 

12 

49 

2 

44 

82 

936355 

4 

26 

068645 

II 

5o 

8 1 5485 

3 

43 

82 

986610 

4 

26 

068800 
io-o63i34 

10 

5i 

9-8i563i 

3 

43 

9.8787661 

82 

9-986866 

4 

25 

9 

52 

815778 

3 

43 

878656ii 

82 

987121 

4 

25 

062879J  8 

53 

815924 

3 

43 

878547!  I 

82 

987876 

4 

35 

0626241  7 
062868  6 

54 

816069 
816215 

3 

43 

878438  I 

82 

987682 

4 

25 

55 

3 

43 

87832811 

82 

987887 

4 

25 

0621 i3   ' 

5 

56 

8i636i 

3 

43 

878219  I 

83 

988142 

4 

35 

06 1 858 

4 

U 

8i65o7 

3 

43 

878109 

I 

83 

938898 
938653 

4 

25 

061602 

3 

8 16652 

3 

43 

877999 

I 

83 

4 

25 

061847 

2 

59 

816798 

3 

43 

877850  I 

83 

988908 

4 

25 

061092 
060837 

I 

66 

816943 

2 

42 

8777go  I 

83 

989168 

4  35 

0 
M. 

Cosine 

D. 

Sine 

49^ 

Cotang. 

D. 

Tang. 

HlSRi 

AND  TANGKNTb. 

(41  DEGREES. 

1 

60 

M.'l 
c 

r-SineH 

T). 

CoBUie  1  D.   Tang. 

D. 

"Oomng. 

"6^" 

9.816943 

2.42 

9.8777801.83  9-939163 

10-060837 

1 

817088 

42 

877670  I 

•83!   939418 

o6o582 

U 

a 

817233 

42 

877560  1 

.83!   939673 

060327 

3 

817379 

42 

877450  I 

-83l   939928 

060072 

57 

4 

817524 

41 

877340  1 

•83 

94018^ 

059817 

56 

5 

817668 

41 

877230  1 

-84 

940438 

359562 

55 

6 

817813 

41 

877120  I 

•84 

940694 

059306 

54 

I 

817958 

41 

877010I1 

•84 

940949 

039001 
058796 

53 

8i8io3 

41 

876899  I 

.84 

941204 

32 

9 

818247 

41 

8766781 

.84 

941458 

058342 

5i 

lO 

818392 
9.818536 

41 

.84 

941714 

o5Sj86 

5o 

11 

40 

9.876068  I 

.84 

9^941968 

10 -05803 2 

'^ 

la 

818681 

40 

8764571 

•84 

942223 

tl]l]l 

i3 

818825 

40 

876347;  I 

■84 

942478 

% 

i4 

818960 
81911J 

40 

876236,1 

.85 

942733 

057267 

i5 

40 

876125  1 

•85 

942988 

057012 

45 

i6 

819257 

40 

876014  1 

.85 

943243 

056757 
o565o2 

44 

\l 

819401 

40 

875904  1 

.85j   94349^ 
.85!   943752 

43 

819545 

39 

875703  1 
875682  I 

056248 

42 

19 

819689 

39 

•85   944007 

n^l 

41 

20 

819832 

39 

875571'! 

-85   944262 

40 

21 

9.819976 

39 

Q.87545Q  1 

•85  9-944517 

10.055483 

^ 

22 

820120 

39 

875348,1 

•85   94i77» 

055229 

23 

820263 

39 

875237:1 

-85   945026 

054974 

U 

24 

820406 

It 

875126  I 

•86   945281 

054719 

25 

82o55o 

875014  I 

•86   945535 

054463 

35 

26 

820693 
820836 

38 

874903  I 

-86   945790 

054210 

34 

u 

3 

38 

874791  1 
874680  I 

.86   946045 

053955 

33 

820979 

_ 

38 

-86   946299 

053701 

32 

29 

821122 

38 

874568  I 

•86   9465D4 

053446 

3i 

3o 

821265 

38 

874456  I 

•86   946808 

053192 

3o 

3i 

9-821407 

38 

9.87/J44  1 

•86  9-947063 

10-052937 
052682 

29 

32 

82i55o 

38 

874232  1 

•87   947318 

28 

33 

821693 

37 

87412111 

•87 

94757a 

052428 

'^ 

34 

821835 

37 

874009  I 

•87 

947826 

052174 

35 

821977 

37 

8738^  I 

•87 

94^°^' 

051919 

25 

36 

822120 

37 

8737841 
87367211 

•87 

948336 

o5i664 

24 

U 

822262 

37 

•87 

948590 

o5i4io 

23 

822404 

37 

873560!  1 

•  87 

948844 

o5ii56 

22 

39 

822546 

11 

873448!  I 

•87 

949099 
9493d3 

030901 

21 

40 

8226S8 

873335  I 

•87 

o5o647 

20 

41 

9.822880 

36 

9.873223,1 

•87 

9  949607 

10 -050393 
o5oi38 

\l 

42 

822972 

36 

8731101 

-88 

949862 

43 

823114 

36 

Z^t;. 

•  88 

950116 

049884 

\l 

44 

823255 

36 

•  88 

950370 

049630 

45 

823307 
823539 

36 

8727721 
8726591 

^88 

950625 

049375 

i5 

46 

36 

•  88 

&^ 

049121 
048867 

14 

a 

823680 

35 

872547:1 

•88 

i3 

823821 

35 

8724341 

-88!   931388 

4 

048612 

12 

49 

823963 

35 

872321  1 

•881   951642 

048358'  11  1 

56 

824104 

35 

872208;! 

•88   951896 
•80'  9-952i5o 

048104 

10 

5i 

9-824245 

35 

9.8720951 

10-047850 

I 

52 

824386 

35 

l]\& 

•8^f 

952405 

047595 

53 

824527 

35 

•  8? 

95265o 
95291J 

047341 

54 

824668 

34 

871755;! 

.8^ 

047087 
046833 

55 

824808 

34 

871641I1 

•8^ 

953167 

56 

824949 

34 

871528,1 

•8? 

953421 

046579 
046325 

ll 

825090 
825230 

34 

871414  1 

■si 

953675 

34 

871301  I 

•89 

953929 
954183 

046071 

59 

825371 

34 

871187,1 

•89 

045817 
045563 

6o 

8255II 

34 

871073,1 

•90 

954437 

(V.sine 

D. 

8iiie  |48c 

Cotaiig. 

I). 

Tatl^. 

tL  1 

GO 

(42 

DEOREKS.)   A  TABLE  OF  LOOARITHMIC 

0 

Sine 

D. 

Cosine  \   D.  |  Tanof. 

I). 

Cotanor. 

9-8255ii 

2.34 

9-871073  I. 90I  9-954487 

4.23 

!0. 045563 

60 

I 

825651 

83 

870960:1 

870846;! 

•90   954691 

4-23 

oi^3o9 
o45o55 

3 

825791 
8a593i 

33 

-90!   934945 

4-23 

3 

33 

87078211 

•90 

955200 

4-23 

044800 

h 

4 

816071 

83 

870618;! 

.9c 

955454 

4-23 

044546 

5 

t' 762 1 1 

33 

870504  I 

•  90 

955707 

4-23 

044298 
044089 

048785 

55 

6 

826351 

33 

870890  ! 

•  90 

955961 

.4-23 

54 

I 

826491 

33 

870276!! 

.90 

956215 

4-23 

53 

826681 

83 

870161  I 

-90 

956469 

4.23 

04353! 

52 

9 

826770 

32 

870047  |l 
869983 j  I 

•91 

95672J 

4-23 

048277 
048028 

5i 

10 

826910 

32 

.91 

956977 

4-23 

5o 

II 

9.827049 

32 

9.869818  I 

•91 

9"9i723i 

4-23 

10-042769 

tl 

la 

827189 
827828 

82 

8697041 

•91 

957485 

4-28 

0425lD 

i3 

32 

8695891 

.9! 

957780 
957998 

4-23 

042261 

47 

14 

827467 

32 

869474I1 

.9! 

4-28 

042007 

46 

i5 

827606 

32 

869860 1 1 

.9! 

958246 

4-23 

041754 

45 

i6 

827745 

82 

86924511 

•91 

958500 

4-23 

o4i5oo 

44 

\l 

827884 

81 

8691801 

-91 

958754 

4.28 

041246 

43 

828023 

81 

8690151 
8689001 

•92 

959008 

4-23 

0.10992 

42 

19 

828 1 62 

3i 

-92 

959262 

4-23 

010788 

41 

20 

828801 

81 

868785]  I 

.92 

959516 

4-28 

040484 

40 

21 

9-828489 
828578 

81 

9-868670  I 

-92 

9-959769 
960023 

4-23 

10-040281 

It 

22 

3i 

868555  I 

.92 

4.28 

089977 

23 

828716 

81 

868440  I 

.92 

960277 

4-28 

089728 

37 

24 

828855 

3o 

868324  I 

.92 

960531 

4-23 

089469 

36 

25 

828998 

80 

868209  1 
868098  ! 

.92 

060784 

4-23 

089216 

35 

26 

829181 

3o 

.92 

961088 

4-23 

088962 

34 

11 

829269 

3o 

867978  I 
867862  1 

•93 

961291 

4.28 

088709 
038455 

33 

829407 

80 

.98 

961545 

4-23 

32 

29 

829545 

3o 

867747  I 

■93 

961799 
962052 

4.28 

088201 

8! 

3o 

829688 

80 

867681  I 

.98 

4-23 

087948 

3o 

3i 

9.829821 

29 

9-8675i5  I 

.98 

9-962806 

4-23 

10-037694 

It 

32 

829959 

29 

867809  I 
867283  I 

-98 

962560 

4-23 

087440 

33 

880097 
880284 

29 

.98 

962818 

4-23 

087187 

11 

34 

29 

867167  I 

.93 

968067 

4-23 

086933 

35 

880872 

29 

867051  I 

-93 

968820 

4-23 

086680 

25 

36 

83o5o9 

29 

866935  I 
866819  I 
866708  1 

.94 

968574 

4.28 

086426 

24 

U 

880646 

29 

.94 

968827 

4-23 

086178 

23 

880784 

It 

•94 

964081 

4-23 

035919 

22 

39 

880921 

866586  I 

•94 

964335 

4-23 

035665 

21 

40 

88io58 

28 

866470  I 

•94 

964588 

4-22 

085412 

20 

41 

9.881195 
881882 

28 

9-866353  i 

.94 

9-964842 

4-22 

io.o35!58 

\t 

42 

28 

866287  1 

.94 

965095 

4-22 

084905 

43 

881469 

28 

866120  I 

.94 

965849 

4-22 

08465! 

17 
!6 

44 

881606 

28 

866004  1 

.95 

965602 

4-22 

084898 

45 

881742 

28 

865887  1 

-95 

965855 

4-22 

084145 

i5 

46 

881879 
832015 

28 

865770  I 

-95 

966105 

4-22 

08889! 

14 

47 

27 

8656531! 

-95 

966862 

4-22 

088688 

i3 

48 

882152 

27 

865586  I 

.95 

966616 

4-22 

038884 

12 

49 

832288 

27 

865419  I 

-95 

Z^i 

4-22 

o83i3i 

II 

5o 

832425 

27 

86530?  I 

.95 

4-22 

082877 

ID 

5i 

^  882561 

27 

9-865i85|r 

.95  9-96737^ 

4-22 

IP  082624 

0 

53 

882697 

0 

• 

27 

865o68|i 

-95   967629 

4-23 

082371 

8 

53 

882888 

11 

864950 j! 
864833! 

.95 

967888 
968186 

4-22 

032!I7 

I 

54 

882969 

.96 

4-22 

081864 

55 

838ioD 

26 

8647161 

.96 

968889 

4-22 

08161! 

5 

56 

«3824i 

26 

864598  I 

.96 

968643 

4.22 

o3i857 

4 

U 

833377 

26 

864481  ! 

-96'   968896 

4-22 

o3iio4 

3 

833512 

26 

864868  I 

-96   969140 
.96   969408 

4-22 

o3o85i 

2  ; 

59 

833648 

26 

864245  I 

4-22 

o3o597 

I 

6o 

833783 

26 

864127  I 

-96   969656 

4-22 

o3o344 
Tarur. 

0 

CI<«iae 

D. 

Sine  kto!  Cotane.  1 

D. 

SINES 

AKD    TANGENTS. 

(43  DEOIiEES.' 

01 

0 

Sine 

D. 

Cosine  |  D, 

Tansr. 

D. 

Cotane. 

~6r 

9.833733 

2.26 

9-864127  1.96 

9.969656 

4-22 

ioo3o3i4 

I 

833019 

• 

25 

664010  1 

.96 

969909 

4 

22 

03009! j  5q 
029838  5& 

1 

834054 

25 

863892  1 

-97 

970162 

4 

22 

3 
4 

834189 
834325 

2 

25 

25 

863774  I 
863656  I 

•97 
-97 

970416 
970669 

i 

22 
22 

l^^\  u 

5 

834460 

2 

25 

863538  I 

■97 

970922 

4 

22 

0290781  55 
028825  5i 

6 

834595 
834730 
834»65 

25' 

863419  I 

-97 
•97 

971175 

4 

2? 

I 

25 

863301  I 

971429 

4 

22 

028571  53 

25 

863 1 83  1 

•97 

9716R2 

4 

22 

0283 18  52 

9 

834999 

24 

863o64  I 

•97 

971935 

4 

22 

0280651  5 1 

10 

835i34 

24 

862946  I 
9-862827,1 

.98 

972188 

4 

2: 

027fii2!  5o 

II 

9.835269 

^  835403 

24 

.98 

9972441 

4 

22 

10^027559  4q 
027306;  48 

12 

24 

862709  1 

.98 

972694 

4 

22 

i3 

835538 

24 

8625901 

.98 

972948 

4 

22 

027052,  47 

14 

835672 

24 

862471  I 

.98 

973201 

4 

22 

026799'  46 

15 

835807 

24 

862353  I 

.98 

973454 

4 

22 

026546  45 

|6 

835941 

24 

862234  I 

.98 

973707 

4 

22 

026293,  44 

',1 

836075 

23 

862115  1 

.98 

973960 

4 

22 

026040!  43 

836209 
836343 

23 

861996  I 
861877  I 

.98 

974213 

4 

22 

0257871  42 

'9 

23 

.98 

974466 

4 

22 

025534;  41 

20 

836477 

23 

861758  I 

.99 

974719 
9-974973 

4 

22 

025281  40 

31 

9.83661 1 

23 

9-86i63S  I 

.99 

4 

22 

10-025027  39 

22 

836745 

23 

86i5i9  1 

.99 

975226 

4 

22 

024774;  38 

23 

836S78 

23 

861400  I 

.99 

975479 

4 

22 

024521'  37 

24 

837012 

22 

861280  1 

•99 

975732 

4 

22 

024268  36 

25 

837140 

22 

861 161  I 

-99 

9759S5 

4 

22 

024015  35 

26 

837279 

22 

861041  1 

.99 

976238 

4 

22 

023762;  34 

11 

83i4i2 

22 

860022' I 
860^02  I 

.99 

976491 

4 

22 

023509  33 

■  837546 

* 

22 

.99 

976744 

4 

22 

023256  32 

29 

837679 

2 

22 

860682 '2 

.00 

976997 

4 

22 

O23oo3|  3i 

3o 

837812 

2 

22 

86o562,2 

.00 

97725o 

4 

22 

022750,  3o 

3i 

g. 837945 

- 

22 

9-8604422 

•00 

9 -977503 

4 

22 

10  022497  20 

32 

83S078 

21 

866322  2 

•00 

977756 

4 

22 

022244  28 

33 

838211 

21 

860202  2 

•00 

978009 

4 

22 

0219911  27 

021738  26 

34 

838344 

21 

860082:2 

.00 

978262 

4 

22 

35 

838477 

21 

859^4212 

.00 

9785 1 5 

4 

22 

021485  25 

36 

838610 

21 

.00 

978768 

4 

22 

021232  24 

ll 

838742 

21 

859721  2 
8596012 

•01 

979021 

4 

22 

020979!  23 

020726}  22 

838875 

21 

•01 

979274 

4 

22 

39 

839007 

21 

8594802 

•01 

979527 

4 

22 

020473 i  21 

4c 

839140 

20 

8593602 

•01 

979780 

4 

22 

020220,  20 

41 

9.839272 

20 

9-8592392 

•01 

9-980033 

4 

22 

10-019967  10 

019714,  18 

42 

839404 

20 

8591 19'2 

•01 

980286 

4 

22 

43 

83q536 

20 

858998.2 
858877  2 

-01 

980538 

4 

22 

019462;  17 
01920-)!  16 

018956!  1 5 

44 

839668 

20 

•01 

980791 

4 

21 

45 

839800 

20 

8587562 

•02 

981044 

4 

21 

46 

839932 

20 

858635  2 

•02 

981297 

4 

21 

018703,  14 

s 

840064 

'9 

8585i4'2 

•02 

981550 

4 

21 

018450  i3 

840196 

"9 

858393  2 

•02 

981803 

4 

21 

018197  12 

49 

84032S 

»9 

858272,2 

•02 

982056 

4 

21 

01 7044  II 

5o 

84045Q 

'9 

858i5i2 

•02 

982309 

4 

21 

017601  r. 

5i 

9.840591 

2 

19 

9-858029  2 
857908,2 

•02 

9-9^2562 

4 

21 

io^oi74J8  9 
017186  r 

52 

840722 

19 

.02 

982814 

4 

21 

53 

840854 

19 

857786*2 

•02 

983067 

4 

21 

016933  7 
016680  6 

54 

8^0985 

It 

85766512 

•  03 

983320 

4 

21 

55 

841116 

85754312 

•  o3 

9S3573 

4 

21 

oi64i7i  5 

56 

841247 
841378 

18 

857422;2 

.o3 

983826 

4 

21 

016174  4 

u 

18 

8573oo'2 

.o3 

984079 

4 

21 

01 592 1   3 
015669   2 

841509 

18 

857178,2 

03 

984331 

4 

21 

59 

841640 

18 

857o56l2 

.03 

984584 

4 

21 

oi54i6|  I 

^ 

84177' 

18 

85693412 

•03 

984837 

4 

21 

oifi63 

0 

CoAino 

D. 

Sii-  Ueo 

Cotanfr. 

D. 

Tail?. 

M^ 

62 

(44 

[    DEGREES.)   A 

TABLE  OF  LOGARITHMIC 

k. 

Sine 

1   D.    -   - 

Cosine  T).       Tang. 

] 

D.   )  Cotang. 

0 

"9^841771" 

1  2-18 

9-856934  2-03,  9-9S4837 

4 

•21   io.oi5i63 

60 

I 

841902 

2 

-18 

8568 1 2  2-03;   985090 

4 

-21      OI49IO 

^ 

2 

842033 

2 

-18 

856690!  2 -04 

985343 

4 

•21      014657 

3 

842163 

2 

•«7 

856568:2-04 

985596 

4 

-21      014404 

u 

4 

842294 

2 

•17 

85644612 -04 

985848 

4 

-21      0l4l52 

5 

842424 

2 

•17 

856323!2-o4 

986 1 01 

4 

-21      013899 

55 

6 

842555 

2 

•n 

856201 

2 -04 

986354 

4 

-21     0136461  54  1 

I 

842685 

2 

•17 

856078 

2-04 

986607 

4 

-21      013393 

53 

842815 

2 

•17 

855956 

2-04 

986860 

4 

-21     oi3i4o 

52 

9 

842946 

2 

•17 

855833 

2-o4i   987112 

4 

•21      012888 

5i 

10 

843076 

2 

•17 

8557II 

2-05   9-^7365 

4 

-21     012635 

5o 

II 

9-843206 

2 

.16 

9-85558Bi2-o5!  9-987618 

4 

-21    10-012382 

it 

12 

843336 

2 

.16 

855465 

2-o5l   9B7871 

4 

-21      O12129 

i3 

843466 

2 

.16 

855? <2 

2-o5 

988123 

4 

21      011877 

47 

14 

843595 

2 

.16 

855219 

2-o5 

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